Type IV Self-Dual Codes over Rings Steven T. Dougherty∗, Philippe Gaborit†, Masaaki Harada‡, Akihiro Munemasa§and Patrick Sol´e¶ June 22, 2011
Keywords: Self-dual codes, divisible codes, codes over rings, lattices Abstract In this paper, we study Type IV self-dual codes over the commutative rings of order 4. Gleason type theorems of Type IV codes and their shadow codes are investigated. A mass formula of Type IV codes over these rings are given. We give a classification of Type IV codes over Z4 and F2 + uF2 for reasonable lengths. We also construct a number of optimal Type IV codes.
1
Introduction
A formally self-dual code (that is, the code and the dual code have identical Hamming weight enumerators) is divisible if there exists a positive integer δ > 1 such that δ divides all non-zero weights in the code. The Gleason-Pierce theorem characterizes the fields for which there exists formally self-dual divisible codes over a finite field Fq of order q. Theorem 1.1 (Gleason-Pierce) Suppose C is a formally self-dual divisible code of length n over Fq , and let δ be the largest positive integer dividing all non-zero weights of C. Then one of the following holds: Type I: Type II: Type III:
q = 2 and δ = 2, q = 2 and δ = 4, q = 3 and δ = 3,
∗
Department of Mathematics, University of Scranton, Scranton, PA 18510, USA INRIA-Dpt Mathematics, University of Illinois–Chicago, 851 S. Morgan, Chicago, IL 60607-7045, USA ‡ Department of Mathematical Sciences, Yamagata University, Yamagata 990–8560, Japan § Graduate School of Mathematics, Kyushu University, Fukuoka 812–8581, Japan ¶ CNRS, I3S, ESSI, BP 145, Route des Colles, 06 903 Sophia Antipolis, France †
1
Type IV: Type V:
q = 4 and δ = 2 or q is arbitrary, δ = 2 and WC (x, y) = (x2 + (q − 1)y 2 )n/2 .
Remark. By a proof given in [32], the above theorem can be applied to codes over finite commutative rings for which the MacWilliams relations hold, for example, to codes over Frobenius rings [34]. Since the MacWilliams relation holds for all finite rings of order 4, any self-dual divisible code over a ring of order 4 which is not Type V is necessarily Type IV. In this paper we generalize Type IV self-dual codes over F4 to codes over three different commutative rings Z4 , F2 + uF2 = {0, 1, u, 1 + u} with u2 = 0 and F2 + vF2 = {0, 1, v, 1 + v} with v 2 = v, of order 4. F2 + vF2 is abstractly isomorphic to F2 × F2 . We shall consider the ring using whichever form is more convenient. Together with F4 these comprise all commutative rings of order 4. Self-dual codes over F2 + uF2 are studied in [1], [2] and [10], and self-dual codes over F2 + vF2 are studied in [1] and [13]. Self-dual codes over Z4 are widely studied, and are especially interesting because of their connection to binary codes via the Gray map, and their connection to real unimodular lattices (cf., e.g. [3], [4], [5], [8, 3rd. ed.], [18] and [20]). There is also an interesting connection to designs [17] and [19]. Likewise, self-dual codes over F2 + uF2 have a connection to unimodular complex lattices [1], [2] and [10]. Self-dual codes over F2 + vF2 are related to 7-modular lattices and a special kind of quantum codes [1] and [13]. This paper is organized as follows. Section 2 gives definitions, basic properties of Type IV codes and families of Type IV codes over Z4 and F2 + uF2 . In Section 3, we study self-dual codes and Type IV codes over F2 + vF2 using the Chinese remainder theorem. In Section 4, we investigate symmetrized and Hamming weight enumerators of Type IV codes. Using invariant theory, Gleason type theorems are derived. The symmetrized weight enumerators of shadow codes of Type IV codes over Z4 and F2 + uF2 are also studied. Section 5 gives a mass formula for Type IV codes over Z4 , F2 + uF2 and F2 + vF2 . In Section 6, a classification of Type IV codes over Z4 and F2 + uF2 are given for reasonable lengths. In Section 7, we determine the highest possible minimum weights using Gleason type theorems given in Section 4. We also construct a number of optimal Type IV codes over Z4 and F2 + uF2 (that is, Type IV codes with highest minimum weight for that length).
2 2.1
Preliminaries Codes over Rings of Order 4
Let R be either Z4 , F2 + uF2 or F2 + vF2 . Throughout this paper if the statement does not depend on which ring we are using we shall denote the ring by R. A code C of length n over R (or an R-code of length n) is an R-submodule of Rn . An element of C is called a 2
codeword of C. A generator matrix of C is a matrix whose rows generate C. We consider different weights for codewords over these rings. The Hamming weight of a codeword is the number of non-zero components. The Euclidean weights for the elements of Z4 are 0, 1, 4 and 1 respectively and for the elements of F2 + uF2 = {0, 1, u, 1 + u}, the Euclidean weights are 0, 1, 4 and 1. The Euclidean weight of a codeword is the rational sum of the Euclidean weights of its components. The Lee weights of the elements of Z4 are 0, 1, 2 and 1 respectively, 0, 1, 2 and 1 for F2 + uF2 , and 0, 2, 1, 1 for F2 + vF2 = {0, 1, v, 1 + v}. The Lee weight of a codeword is the rational sum of the Lee weights of its components. The Hamming and Lee distances between two codewords x and y are the Hamming and Lee weights of x − y, respectively. For the ring F2 + vF2 , another weight (we call it the Bachoc weight) is defined in [1]. The Bachoc weights are 0, 1, 2, and 2 for {0, 1, v, 1 + v}. The equivalence with the definition in [1] comes from the ring isomorphism a + vb 7→ (a + b)v + a(1 + v) which maps F2 + vF2 onto F2 × F2 . The minimum Hamming, Euclidean, Lee and Bachoc weights, dH , dE , dL and dB of C are the smallest Hamming, Euclidean, Lee and Bachoc weights among all non-zero codewords of C, respectively. We consider the following rings and maps. Fx22
F2 + uF2 −−−−−→ ψ
β F2 + vF2
←−−−−− Z4 φ
ψ ψ(0) = 00 ψ(1) = 01 ψ(1 + u) = 10 ψ(u) = 11
φ φ(0) = 00 φ(1) = 01 φ(3) = 10 φ(2) = 11
β β(0) = 00 β(1) = 11 β(1 + v) = 10 β(v) = 01
The maps ψ, φ and β are isometries from (R, Lee distance) to (F22 , Hamming distance), and are called Gray maps. These maps are extended to Rn naturally. The maps ψ and β are linear, but φ is not. Let x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) be two elements of Rn . We define the inner product of x and y in Rn by x·y = x1 y1 +· · ·+xn yn where the operations are performed in R. For codes over F2 + vF2 we consider two inner products, namely the Euclidean inner product P P given by xi yi and the Hermitian inner product given by xi yi where 0 = 0, 1 = 1, v = v+1 and v + 1 = v. The dual code C ⊥ of C is defined as C ⊥ = {x ∈ Rn | x · y = 0 for all y ∈ C}. C is self-dual if C = C ⊥ . Note that self-dual codes exist for all n > 0 for both codes over Z4 and F2 + uF2 since 2 and u generate self-dual codes of length 1. Self-dual codes exist only for even lengths over F2 + vF2 for the Euclidean inner product, but they exist for all lengths with the Hermitian inner product since v generates a self-dual code of length 1. In this paper, codes with respect to the Euclidean (resp. Hermitian) inner product are shortly said to be Euclidean (Hermitian) codes. 3
We say that two codes are equivalent if one can be obtained from the other by permuting the coordinates and (if necessary) interchanging the two elements 1 and 3 of certain coordinates for R = Z4 and the two elements 1 and 1 + u of certain coordinates for R = F2 + uF2 . Codes differing by only a permutation of coordinates are called permutation-equivalent. For R = F2 + vF2 , we say that C and C 0 are equivalent if either C and C 0 are permutationequivalent or C is permutation equivalent to the code obtained from C 0 by changing v and 1 + v in all coordinates. For R = Z4 and F2 + uF2 , the automorphism group Aut(C) of C consists of all permutations and changes of the above two elements of the coordinates that preserve C. Several weight enumerators are associated with a code over R. In this paper, we deal with the symmetrized weight enumerators. The symmetrized weight enumerator (swe) of a code C over R is given by sweC (a, b, c) =
X
an0 (x) bn1 (x)+n3 (x) cn2 (x) ,
x∈C
where ni (x) is the number of components of x ∈ C that are i in Z4 , and n0 (x), n1 (x), n2 (x) and n3 (x) are the number of components of x ∈ C that are 0, 1, u and 1 + u respectively in F2 + uF2 . For codes over F2 + vF2 we define the swe by sweC (a, b, c) =
X
an0 (x) bn1 (x) cn2 (x) ,
x∈C
where ni (x) is the number of components of x whose Lee weight is i. Klemm [21] established the MacWilliams identities for a code over Z4 . Wood [34] established the MacWilliams identities for codes over any finite Frobenius ring. For codes over F2 + vF2 the MacWilliams relations for the swe are the same for both inner products [13]. Theorem 2.1 (Klemm [21], Wood [34]) For a code C over a commutative ring of order 4 we have: 1 sweC ⊥ (a, b, c) = sweC (a + 2b + c, a − c, a − 2b + c). |C| There is a remarkable class (called Type II) of self-dual codes over Z4 and F2 + uF2 satisfying some divisibility condition with respect to Euclidean and Lee weights, respectively. Type II codes over Z4 were first defined in [4] as self-dual codes containing the all-ones vector and with the property that all Euclidean weights are divisible by eight. However, recently it has been shown in [20] that any self-dual code with the property that all Euclidean weights are divisible by eight is equivalent to a Type II code. Thus in this paper, we say that selfdual codes over Z4 with the property that all Euclidean weights are divisible by eight are Type II. For codes over F2 +uF2 we say that the code is Type II if all the Lee weights are multiples of 4 [10]. Note that we could have used the same definition for codes over F2 + uF2 as we 4
did for Z4 , however it will become clear why this definition is made after examining the images of the codes under the Gray map and the lattices formed by the codes. Self-dual codes which are not Type II are called Type I. Self-dual codes over R with even Hamming weights will be called hence-forth Type IV. We shall adopt the following notation. If a code is a Type IV code then we shall denote it as a Type IV-I (resp. Type IV-II) if it is also a Type I (resp. Type II) code.
2.2
Residue Codes, Torsion Codes and Binary Codes
Any code over Z4 is permutation-equivalent to a code C with generator matrix of the form
(1)
I A B1 + 2B2 k1 , 0 2Ik2 2D
where A, B1 , B2 and D are (1, 0)-matrices. We say that a code with generator matrix (1) has type 4k1 2k2 [9]. The binary [n, k1 ] code C (1) with generator matrix (2)
Ik1 A B1
,
is called the residue code of the Z4 -code. The binary [n, k1 + k2 ] code C (2) with generator matrix I A B k 1 1 , (3) 0 Ik2 D is called the torsion code of the Z4 -code. ⊥ If C is self-dual then C (1) is doubly-even and C (2) = C (1) [9]. Lemma 2.2 If C is a Type IV code over Z4 then the residue code C (1) contains the all-ones vector 1. Proof. Since C is Type IV, the number of 2’s in each codeword of C is even. Thus the ⊥ torsion code C (2) is even and C (1) =C (2) contains 1. 2
Proposition 2.3 A Type IV code C over Z4 is Type IV-II if and only if all the Hamming weights of C (1) are multiple of 8. Proof. Suppose that C is Type IV. Let x be a codeword of C. Since C is self-dual, n1 (x) + n3 (x) ≡ 0 (mod 4). Since C is Type IV, n2 (x) is even. C is Type II if and only if n1 (x) + n3 (x) ≡ 0 (mod 8). 2
5
For F2 + uF2 any code is permutation-equivalent to a code C with generator matrix of the form I A B + uB k 1 2 1 , (4) 0 uIk2 uD where A, B1 , B2 and D are matrices over F2 . We say that a code with generator matrix (4) has type 4k1 2k2 . The binary [n, k1 ] code C (1) with generator matrix
(5)
Ik1 A B1
,
is called the residue code of the code. The binary [n, k1 + k2 ] code C (2) with generator matrix
(6)
Ik1 A B1 , 0 Ik2 D
is called the torsion code of the F2 + uF2 -code. ⊥ If C is self-dual then C (1) is self-orthogonal and C (2) = C (1) [10]. Similarly to Lemma 2.2, we have the following: Lemma 2.4 If C is a Type IV code over F2 + uF2 then the residue code C (1) contains the all-ones vector 1. Proposition 2.5 A Type IV code C over F2 + uF2 is Type IV-II if and only if C (1) is doubly-even. Proof. Let x be a codeword of C. Since C is self-dual, n1 (x) + n3 (x) ≡ 0 (mod 2). Since C is Type IV, n2 (x) is even. C is Type II if and only if n1 (x) + n3 (x) ≡ 0 (mod 4). 2
2.3
Properties of Type IV Codes
Proposition 2.6 If C is a Type IV Z4 -code of length n then all the Lee weights of C are divisible by four and its Gray image φ(C) is a self-dual Type II binary code. Proof. Let x be an arbitrary codeword of C. By hypothesis n1 (x) + n2 (x) + n3 (x) is even, and by self-duality n1 (x) + n3 (x) is a multiple of 4. Hence n2 (x) is even, and the Lee weight n1 (x) + 2n2 (x) + n3 (x) is a multiple of 4. By Theorem 8 in [5], φ(C) is linear then φ(C) is also a binary Type II self-dual code. 2 As a corollary to Proposition 2.6, we have the following: Corollary 2.7 The minimum Lee weight dL of a Type IV Z4 -code of length n is bounded by dL ≤ 4(1 + bn/12c). 6
Proof. The minimum Hamming weight d of a Type II binary code of length 2n is bounded by d ≤ 4(1 + bn/12c). This bound was proved by Mallows and Sloane [26]. The bound follows immediately from the Mallows and Sloane bound. 2 The binary Gray map image of a Type IV F2 + uF2 -code is a self-dual code but not necessarily a Type II binary code. For example, the code C2 = {(0, 0), (1, 1), (u, u), (1 + u, 1 + u)} is self-dual and has Hamming weight enumerator x2 + 3y 2 . Its binary image is {(0, 0, 0, 0), (0, 1, 0, 1), (1, 1, 1, 1), (1, 0, 1, 0)} which is a Type I code. Corollary 2.8 A Type IV code over Z4 of length n exists if and only if n ≡ 0
(mod 4).
Proof. It is known that there is a Type II binary code of length n if and only if n ≡ 0 (mod 8). Thus a Type IV Z4 -code of length n exists then n ≡ 0 (mod 4). The code D4⊕ in [9] is a Type IV Z4 -code of length 4. 2 As a corollary to Proposition 2.3, we have the following: Corollary 2.9 There is no Type IV-II code of type 4n/2 , where n is the length of the code. Proof. If a Type IV code C is of type 4n/2 then the binary residue code C (1) is a Type II code with all weights divisible by eight. Theorem 1.1 shows that there cannot exist such a code. 2 By the classification given in Section 6, there is no Type IV-I code of type 4n/2 , for length n ≤ 12.
2.4
Infinite Families
First we present infinite families of Type IV codes over Z4 . • Klemm Codes: The Klemm codes Kn of length n = 4m are constructed by a bilevel construction from the repetition code Rn and its dual the parity-check code Pn . Kn := Rn + 2Pn = 2Pn
[
(1 + 2Pn )
where 1 is the all-ones vector. Their symmetrized weight enumerators are 1 sweKn (a, b, c) = ((a + c)n + (a − c)n ) + 2n−1 bn . 2 It is easy to see that the order of Aut(K4m ) is (4m!)24m−1 . Thus it follows from the mass formula in Section 5 that K4m is a unique Type IV code of type 41 24m−2 . 7
• Cm,r : These codes Cm,r were introduced in [4] as bilevel constructions from binary ReedMuller codes. For 3r ≤ m − 1 let Cm,r := RM (r, m) + 2RM (m − r − 1, m). The code Cm,r is a Type IV Z4 -code. Now we construct analogues of the above two families for F2 + uF2 . Proposition 2.10 Let C, D be a dual pair of binary codes with even weights and C ⊆ D. Then C + uD is a Type IV code over F2 + uF2 . Proof. Let c be a codeword of C and let d be a codeword of D. For a codeword x = c + ud of C + uD, as in [4], we have n1 (x) + n3 (x) = w(c) and n2 (x) = w(d) − w(d ∗ c) where w(y) is the Hamming weight of y and d ∗ c = ((d ∗ c)1 , . . . , (d ∗ c)n ) with (d ∗ c)i = di ci . By hypothesis n1 (x) + n3 (x) and w(d) are even. By duality w(d ∗ c) is even. Hence n1 (x) + n2 (x) + n3 (x) is even. 2 By the above proposition, we have the following two classes. • Klemm Codes: The Klemm codes Kn of length n = 2m are constructed by a bilevel construction from the repetition code Rn and its dual the parity-check code Pn . Kn := Rn + uPn . Their symmetrized weight enumerators are 1 sweKn (a, b, c) = ((a + c)n + (a − c)n ) + 2n−1 bn . 2 The order of Aut(K2m ) is (2m!)22m−1 . Thus it follows from the mass formula in Section 5 that K2m is a unique Type IV code of type 41 22m−2 . • Cm,r : For 3r ≤ m − 1 let Cm,r := RM (r, m) + uRM (m − r − 1, m). The code Cm,r is Type IV.
8
3
Self-Dual Codes and Type IV Codes over F2 + v F2
In this section, we study self-dual codes and Type IV codes over F2 + vF2 using the Chinese remainder theorem. First define the map Φ : F2 + vF2 → F2 × F2 , where Φ(0) = (0, 0), Φ(1) = (1, 1), Φ(v) = (0, 1) and Φ(1 + v) = (1, 0). Φ is a ringisomorphism by the Chinese remainder theorem. The map is extended to (F2 + vF2 )n naturally. Let C be a code over F2 + vF2 then there are binary codes C1 and C2 such that C = Φ−1 (C1 , C2 ) and we denote C by CRT (C1 , C2 ). Note that C1 and C2 are uniquely determined for each CRT (C1 , C2 ). Let c be a codeword of C then c can be uniquely written as c = Φ−1 (c1 , c2 ) where c1 and c2 are codewords of C1 and C2 , respectively. Let wH (c), wL (c) and wB (c) be the Hamming, Lee and Bachoc weights of c, respectively. Then
(7)
wH (c) = wH (c1 ) + wH (c2 ) − wH (c1 ∗ c2 ), wL (c) = wH (c1 ) + wH (c2 ), wB (c) = 2wH (c1 ) + 2wH (c2 ) − 3wH (c1 ∗ c2 ).
Lemma 3.1 Let CRT (C1 , C2 ) and CRT (C10 , C20 ) be codes over F2 + vF2 . CRT (C1 , C2 ) and CRT (C10 , C20 ) are equivalent if and only if there exists a permutation which sends (C1 , C2 ) to (C10 , C20 ) or to (C20 , C10 ). Proof. Follows from the definition of the equivalence.
2
Proposition 3.2 (Dougherty et al. [11]) CRT (C1 , C2 ) is a Euclidean self-dual code if and only if C1 and C2 are binary self-dual codes. Corollary 3.3 A Euclidean self-dual code of length n exists if and only if n is even. Proof. Binary self-dual codes exist if and only if the length is even.
2
Corollary 3.4 Let CRT (C1 , C2 ) be a Euclidean self-dual code. CRT (C1 , C2 ) is Type IV if and only if C1 = C2 . Proof. By Proposition 3.2, C1 and C2 are binary self-dual. Thus all codewords of C1 and C2 have even weights. If CRT (C1 , C2 ) is Type IV then wH (c) is even for any codeword c of CRT (C1 , C2 ). By (7), wH (c1 ∗ c2 ) is even. It turns out that C1 = C2⊥ then C1 = C2 . Conversarly if C1 = C2 then the Hamming weight of any codeword of CRT (C1 , C2 ) is even by (7). 2
9
By Lemma 3.1 and the above corollary, the classification of binary self-dual codes determines the classification of Euclidean Type IV codes. Since all binary self-dual codes of length n ≤ 30 were classified (cf. [7]), Euclidean Type IV codes of length up to 30 are classified. Proposition 3.5 (Bachoc [1]) CRT (C1 , C2 ) is a Hermitian self-dual code if and only if C1 = C2⊥ . Corollary 3.6 Let CRT (C1 , C2 ) be a Hermitian self-dual code. CRT (C1 , C2 ) is Type IV if and only if C1 and C2 are even. Proof. Suppose that CRT (C1 , C2 ) is Type IV. By Proposition 3.5, C1 = C2⊥ . It follows from (7) that wH (c1 ) + wH (c2 ) is even for all codewords c1 and c2 in C1 and C2 , respectively. Thus take the zero-vector as c1 then wH (c2 ) is even. Similarly take the zero-vector as c2 then wH (c1 ) is even. Therefore C1 and C2 must be even codes. Conversely if C1 = C2⊥ , C1 and C2 are even then CRT (C1 , C2 ) is Type IV by (7). 2
Corollary 3.7 If C is a Euclidean Type IV code, then C is Hermitian Type IV. Proof. Let C = CRT (C1 , C2 ) then C1 = C2 and C1 is self-dual by Corollary 3.4.
2
Therefore Euclidean Type IV codes are a special class of Hermitian Type IV codes. We now give divisibility conditions of Lee and Bachoc weights for self-dual codes and Type IV codes over F2 + vF2 . Corollary 3.8 Let C be a Euclidean self-dual code. Then the Lee weight of a codeword of C is even. Moreover if C is Type IV then all the Bachoc weights are even. Proof. Follows from Proposition 3.2, Corollary 3.4 and (7).
2
Corollary 3.9 Let C be a Hermitian self-dual code. Then the Bachoc weight of a codeword of C is even. Moreover if C is Type IV then all the Lee weights are even. Proof. Follows from Proposition 3.5, Corollary 3.6 and (7).
2
We now consider the complete weight enumerator of a self-dual code C over F2 + vF2 defined by X cweC (a, b, c, d) = an0 (x) bnv (x) cn1+v (x) dn1 (x) , x∈C
10
where ni (x) denotes the number of components i in x. Of course, cweC (a, b, b, c) = sweC (a, b, c). Let JC,C 0 (a, b, c, d) denote the joint weight enumerator of binary codes C and C 0 (see [24] for the definition of this weight enumerator). JC,C (a, b, c, d) is called the biweight enumerator of C. Let CRT (C1 , C2 ) be a self-dual code over F2 + vF2 . From the definition, it is easy to see that cweCRT (C1 ,C2 ) (a, b, c, d) is the same as JC1 ,C2 (a, b, c, d). Moreover if C1 = C2 (that is, Type IV) then cweCRT (C1 ,C1 ) (a, b, c, d) is the same as JC1 ,C1 (a, b, c, d). A basis for the space of invariants to which the joint weight enumerators and biweight enumerators belong for binary self-dual codes was given in [24]. Hence the rings for complete weight enumerators of Euclidean self-dual codes and Type IV codes are determined.
4
Invariants and Weight Enumerators
4.1
Some Groups of Orders 384 and 48
We let G384 denote the matrix group of order 384 with generators
1 2 1 1 M1 = 1 0 −1 , 2 1 −2 1
1 0 0 1 0 0 M2 = 0 i 0 and M3 = 0 1 0 . 0 0 1 0 0 −1
The group G384 is the unitary reflection group G(4, 1, 3) and has the following Molien series: (1 −
t4 )(1
1 = 1 + t4 + 2 t8 + 3 t12 + 4 t16 + 5 t20 + · · · . − t8 )(1 − t12 )
Let C[a, b, c]G denote the invariant ring of a group G. Lemma 4.1 C[a, b, c]G384 = C[f4 , f8 , f12 ] where fn = 21 ((a + c)n + (a − c)n ) + 2n−1 bn . Proof. f4 , f8 and f12 are invariant under the group G384 , and the Molien series describes the structure of the ring. The result follows. 2 Let G48 be the matrix group of order 48 generated by M1 , M4 and M5 where
1 0 0 1 0 0 M4 = 0 −1 0 and M5 = 0 −1 0 . 0 0 1 0 0 −1 11
The group G48 is a subgroup of G384 isomorphic to the Coxeter group of type B3 . The Molien series of G48 is (1 −
t2 )(1
1 = 1 + t2 + 2t4 + 3t6 + 4t8 + 5t10 + · · · . − t4 )(1 − t6 )
Similarly to the above lemma, we easily have the following: Lemma 4.2 C[a, b, c]G48 = C[f2 , f4 , f6 ].
4.2
Invariants for Type IV Z4 -Codes
The symmetrized weight enumerator of a self-dual Z4 -code is invariant under the MacWilliams identity, and is therefore invariant under the matrix M1 . By the proof of Proposition 2.6, a codeword x of a Type IV code satisfies the condition that n1 (x) + n3 (x) ≡ 0 (mod 4) and n2 (x) is even. Thus the symmetrized weight enumerator of a Type IV code is invariant under the matrices M2 and M3 . By Lemma 4.1, we have the following characterization of symmetrized weight enumerator of a Type IV code. Theorem 4.3 The symmetrized weight enumerator of a Type IV-I code over Z4 belongs to the ring C[f4 , f8 , f12 ]. The ring is generated by symmetrized weight enumerators of Type IV codes. Remark. fn is the symmetrized weight enumerator of the Klemm code Kn . The ring corresponding to swe’s for Type IV-II codes was determined in [5].
4.3
Invariants for Type IV F2 + uF2 -Codes
Theorem 4.4 The symmetrized weight enumerator of a Type IV-I F2 + uF2 -code belongs to the ring C[f2 , f4 , f6 ]. The ring is generated by the symmetrized weight enumerators of Type IV codes over F2 + uF2 . Proof. M4 and M5 correspond to the conditions that n2 (x) ≡ 0 (mod 2) and the Hamming weight of x is even, respectively, for a codeword x of a Type IV code. Thus the symmetrized weight enumerator of a Type IV code is invariant under the group generated by three matrices M1 , M4 and M5 . The invariant ring for the group follows from Lemma 4.2. The second assertion follows from the fact that fn is the swe of the Klemm code Kn . 2
Corollary 4.5 A Type IV F2 + uF2 -code of length n exists if and only if n is even. 12
Proof. The above theorem gives that if a Type IV code of length n exists then n must be even. This shows that this condition is necessary. The code K2 is a Type IV code of length 2. Taking direct sums of K2 the sufficiency follows. 2
The swe of a Type IV-II F2 + uF2 -code is also held invariant by the matrix:
1 0 0 M6 = 0 i 0 , 0 0 −1 where i is a primitive 4-th root of unity. In other words, the swe of a Type IV-II F2 + uF2 code is invariant under the group G0 generated by M1 , M4 , M5 and M6 . Since M4 = M22 , M5 = M22 M3 and M6 = M2 M3 , we see that G0 = G384 . Lemma 4.1 gives the following: Theorem 4.6 The symmetrized weight enumerator of a Type IV-II F2 + uF2 -code belongs to the ring C[f4 , f8 , f12 ]. The ring is generated by the symmetrized weight enumerator of Type IV-II F2 + uF2 -codes. Corollary 4.7 A Type IV-II F2 +uF2 -code of length n exists if and only if n ≡ 0
(mod 4).
Proof. The above theorem gives that if a Type IV-II code of length n exists then n must be divisible by 4. K4 is a Type IV-II code of length 4. 2
4.4
Invariants for Type IV F2 + vF2 -Codes
At the end of Section 3 we investigated the invariant ring for Euclidean Type IV codes. Thus here we consider only Hermitian Type IV codes. Hermitian self-dual codes are held invariant by the matrices M1 and the matrix:
1 0 0 0 1 0 . 0 0 −1 The symmetrized weight enumerator of a Type IV code is also invariant under the matrix M5 . It is easy to see that the group generated by the above three matrices is the same as G48 . Therefore we have the following characterization. Theorem 4.8 The symmetrized weight enumerator of a Hermitian Type IV F2 + vF2 -code belongs to the ring C[f2 , f4 , f6 ]. The ring is generated by the symmetrized weight enumerators of Type IV-II F2 + vF2 -codes. 13
Proof. It can be easily seen that C[f2 , f4 , f6 ] = C[f2 , f4 , f6 ] where fn is the symmetrized weight enumerator of CRT (Rn , Pn ). The symmetrized weight enumerator of CRT (Rn , Pn ) was computed in [1]. 2
Corollary 4.9 A Hermitian Type IV F2 + vF2 -code of length n exists if and only if n is even. Proof. The above theorem gives that if a Hermitian Type IV code of length n exists then n is even. {(0, 0), (1, 1), (v, v), (1 + v, 1 + v)} is a Type IV code of length 2. 2
4.5
Invariants for Hamming Weight Enumerators of Type IV Codes
The Hamming weight enumerator WC (x, y) of a code C is defined as WC (x, y) = sweC (x, y, y). Let Φ : C[a, b, c] → C[x, y] be the homomorphism of the polynomial rings defined by Φ : f 7→ g(x, y) = f (x, y, y). Then it is easy to see that Φ(C[f4 , f8 , f12 ]) = C[g4 , g12 ] ⊕ g8 C[g4 , g12 ] and Φ(C[f2 , f4 , f6 ]) = C[g2 , g6 ], where gn = Φ(fn ) = 21 ((x + y)n + (x − y)n ) + 2n−1 y n . By Theorems 4.3, 4.4 4.6 and 4.8, we have the following: Corollary 4.10 The Hamming weight enumerator of a Type IV code over Z4 and a Type IVII code over F2 + uF2 belongs to C[g4 , g12 ] ⊕ g8 C[g4 , g12 ], and this is generated by their Hamming weight enumerators. The Hamming weight enumerator of a Type IV-I code over F2 + uF2 and a Type IV code over F2 + vF2 belongs to C[g2 , g6 ], and this is generated by the Hamming weight enumerator of Type IV-I codes over F2 + uF2 . Codes over any two rings with the same cardinality have identical MacWilliams relations for the Hamming weight enumerator. We now compare the rings corresponding to Type IV codes over R with the ring corresponding to Type IV codes over F4 . It was shown in [24] that the Hamming weight enumerator of a Type IV code over F4 belongs to the ring C[x2 + 3y 2 , x6 + 45x2 y 4 + 18y 6 ] = C[x, y]G4 where G4 is the dihedral group of order 12 with generators 1 1 3 1 0 . and 2 1 −1 0 −1 It is easy to see that C[x, y]G4 = C[g2 , g6 ]. A Type IV code over Z4 and a Type IV-II code over F2 + uF2 exists only for lengths n ≡ 0 (mod 4). Thus it is natural to consider the 0 invariant ring C[x, y]G4 where G04 is generated by G4 and the additional matrix diag(i, i). Then it is easy to see that 0
C[x, y]G4 = C[g4 , g12 ] ⊕ g8 C[g4 , g12 ]. 14
The Hamming weight enumerator of a Type IV code over R belongs to C[g2 , g6 ]. Therefore the upper bound on minimum weights of Type IV codes over F4 given in [25] is applied to Type IV codes over R. Corollary 4.11 The minimum Hamming weight of a Type IV code over R of length n is bounded by dH ≤ 2(1 + bn/6c). For instance the Z4 -code D4⊕ of [9] meets the bound with equality (dH = 2). However this bound is not tight in general. Lam and Pless [22] showed that there is no Type IV [24, 12, 10] code over F4 . This existence was a long-standing question. Similarly, we shall show that there is no Type IV code with minimum Hamming weight 10 over Z4 and F2 +uF2 for length 24 in Section 7. Now we investigate the Hamming weight enumerators of the binary self-dual codes obtained from Type IV codes over Z4 and F2 + uF2 by the Gray maps. The Hamming weight enumerator of such a binary self-dual code belongs to the ring Ψ(C[f2 , f4 , f6 ]) where Ψ : C[a, b, c] → C[x, y] is the map defined by Ψ : a 7→ x2 , b 7→ xy and c 7→ x2 . It can be easily shown that Ψ(C[f2 , f4 , f6 ]) = C[Ψ(f2 ), Ψ(f4 )]. Since the length of the self-dual codes is divisible by four, its Hamming weight enumerator belongs to the invariant ring C[x, y]H where H is the group of order 32 generated by
1 1 1 √ , 2 1 −1
i 0 1 0 , and 0 i 0 −1
where i is a primitive complex fourth root of unity. We note that C[x, y]H = C[Ψ(f2 ), Ψ(f4 )].
4.6
The Symmetrized Weight Enumerators of Shadow Codes
The shadow of a self-dual code over Z4 has been introduced in [12]. Namely, S = C0⊥ − C where C0 is the subcode of vectors whose Euclidean weight is a multiple of 8. The swe of the shadow is easily determined by sweS (a, b, c) = sweC (b + ξ(a − c)/2, (a + c)/2, b − ξ(a − c)/2), where ξ is a primitive complex eighth root of unity [12]. Applying this to the swe of the codes K4 , K8 and K12 we have that S4 = 4a3 c+4ac3 +8b4 , S8 = f8 , and S12 = 12a11 c + 220a9 c3 + 792a7 c5 + 792a5 c7 + 220a3 c9 + 12ac11 + 2048b12 . Note that K8 is a Type II code so that its shadow is defined to be the code itself. This gives the following:
15
Corollary 4.12 If the swe of a Type IV Z4 -code of length 4m is X
j , al,j f4m−2l−3j f8l f12
2l+3j≤m
for some complex numbers al,j , then the swe of the shadow is a polynomial in S4 , S8 , S12 of the form X j al,j S4m−2l−3j S8l S12 . 2l+3j≤m
Now let us consider the shadow of a Type I code C over F2 + uF2 . We define C0 as the subcode of C consisting of all codewords of C with Lee weights a multiple of 4. Clearly C0 is of index 2 in C and C of index 2 in C0⊥ . We define the shadow S of C as S = C0 ⊥ − C [10]. Similarly to Z4 , the swe of the shadow is easily determined by sweS (a, b, c) = sweC (b + (a + c)/2, i(a − c)/2, b − (a + c)/2), where i is a complex fourth root of unity [10]. Define the following polynomials: f20 = 2ac + 2b2 , f40 = f4 , f60 = 30ab4 c + 15a2 b2 c2 + (15/2)a4 b2 + (15/2)b2 c4 −(3/2)a5 c + 5a3 c3 − (3/2)ac5 + 2b6 . Using these we have the following corollary. Corollary 4.13 If the swe of a Type IV-I F2 + uF2 -code C can be expressed as X
αjk (f2 )
n−4j−6k 2
(f4 )j (f6 )k ,
n−4j−6k 2
(f40 )j (f60 )k .
j,k
then the swe of its shadow is X
αjk (f20 )
j,k
5
Mass Formulas of Type IV Codes
In this section, we provide mass formulas of Type IV codes over Z4 , F2 + uF2 and F2 + vF2 . To find mass formulas, we first characterize the structure of Type IV codes over Z4 , F2 + uF2 and F2 + vF2 . Let C be a code over R of length n and let x and y be two codewords of C. We denote by ni,j (x, y) the number of occurrences of the couples (i, j) or (j, i) in the columns of the 2 by n matrix obtained from the juxtaposition of x and y. In the following, as there is no ambiguity, we will simply write ni,j for ni,j (x, y). 16
5.1
Mass Formulas of Z4 -Codes
Lemma 5.1 Let x and y be codewords of a self-dual code C over Z4 . Write x = x1 +2x2 , y = y1 + 2y2 , where x1 , x2 , y1 and y2 are (1, 0)-vectors. Then 2(wH (x + y) − wH (x) − wH (y)) ≡ wH (x1 ∗ y1 ) (mod 4), where wH (x) denotes the Hamming weight of x. Proof. From the definition of Hamming weight, wH (x + y) = wH (x) + wH (y) − 2(n2,2 + n1,3 ) − (n1,1 + n1,2 + n2,3 + n3,3 ). Thus (8)
2(wH (x + y) − wH (x) − wH (y)) ≡ 2(n1,1 + n1,3 + n3,3 ) +2(n1,2 + n1,3 + n2,3 + 2n3,3 )
(mod 4).
From the definition of ni,j , (9)
n1,1 + n1,3 + n3,3 = wH (x1 ∗ y1 ),
and (10)
n1,2 + n1,3 + n2,3 + 2n3,3 = wH (x1 ∗ y2 ) + wH (y1 ∗ x2 ).
Since the code C is self-dual, (11)
x · y ≡ wH (x1 ∗ y1 ) + 2(wH (x1 ∗ y2 ) + wH (y1 ∗ x2 )) ≡ 0
(mod 4). 2
Substituting (9)–(11) into (8), we obtain the desired result.
The above lemma allows us to give a complete description of the Type IV Z4 -codes in the following theorem. Theorem 5.2 Let C be a code over Z4 . Suppose that C (1) and C (2) have generator matrices given by (2) and (3), respectively. If C is Type IV, then there exists a unique (1, 0)-matrix B such that I + 2B A B 1 k1 (12) 0 2Ik2 2D is a generator matrix of C. Moreover, we have ⊥
1) C (2) = C (1) , 2) The residue code C (1) contains the all-ones vector, and wH (x ∗ y) ≡ 0 any x and y ∈ C (1) ,
(mod 4) for
3) The number of 2’s in each row of Ik1 + 2B is even, and the matrix B is symmetric.
17
Conversely, if C (1) and C (2) are binary codes with generator matrices given by (2) and (3), respectively, and if the conditions 1)–3) are satisfied, then the Z4 -code C with generator matrix (12) is a Type IV code. Proof. Since C is self-dual, Section VI of [15] implies that there exists a unique matrix B such that (12) is a generator matrix of C, and 1) holds. If C is Type IV then the number of 2’s in every codeword of C is even. By Lemma 2.2, C (1) contains the all-ones vector, and by Lemma 5.1, wH (x∗y) ≡ 0 (mod 4) for any x, y ∈ C (1) . Since ( Ik1 A B1 )( Ik1 A B1 )T ≡ 0 (mod 4) by Lemma 5.1, 0 ≡ ( Ik1 + 2B A B1 )( Ik1 + 2B A B1 )T ≡ 2B + 2B T
(mod 4).
Thus B is symmetric. Conversely, under the conditions 1)–3), the code C is self-dual and, each row of the matrix (12) has even Hamming weight. Lemma 5.1 together with the last part of 2) implies that each codeword of C has even Hamming weight. 2 From the above theorem we deduce a mass formula for Type IV Z4 -codes. We recall that finding a mass formula for Type IV Z4 -codes is equivalent to finding the number of distinct Type IV Z4 -codes of length n [15]. Thus we have the following: Theorem 5.3 Let NdIV (n) be the number of distinct Type IV codes of length n and let τ1 (n, k) be the number of distinct binary codes C 0 of length n and dimension k, containing the all-ones vector, and satisfying the condition wH (x∗y) ≡ 0 (mod 4) for all x and y ∈ C 0 , then X k(k−1) τ1 (n, k) · 21+ 2 . NdIV (n) = k≤ n 2
Proof. Theorem 5.2 implies that any Type IV Z4 -code is completely determined by its residue code, its torsion code and the matrix B. The number of choices for the residue code of dimension k is τ1 (n, k) and the torsion code is determined uniquely by the residue code. It remains to compute the number of choices for B. We can choose freely the diagonal entries in B and the entries below the diagonal except all the entries in the first column. Since B is symmetric, the entries above the diagonal are obtained from the entries below the diagonal. The entries of the first column except the first row are determined by the condition that (k−1)(k−2) 2 the number of 2’s in each row of I + 2B is even. Therefore there are 2k+ ways of choosing B. Note that the number of 2’s in the first row of I + 2B is automatically even, since the number of 2’s in I + 2B except the first row is even and the total number of 2’s in I + 2B is even by the symmetry of B. 2 Remark. We give the values of τ1 (n, k) for small n, which are used in Section 6 in order to check that the classification is complete: 18
• τ1 (4, 0) = 0, τ1 (4, 1) = 1, τ1 (4, 2) = 0, • τ1 (8, 0) = 0, τ1 (8, 1) = 1, τ1 (8, 2) = 35, τ1 (8, 3) = 0, τ1 (8, 4) = 0, • τ1 (12, 0) = 0, τ1 (12, 1) = 1, τ1 (12, 2) = 495, τ1 (12, 3) = 5775, τ1 (12, 4) = 0, τ1 (12, 5) = 0, τ1 (12, 6) = 0.
Corollary 5.4 Let NdIV −II (n) be the number of distinct Type IV-II codes of length n and let τ2 (n, k) be the number of distinct binary self-orthogonal codes C 0 of length n and dimension k, containing the all-ones vector, and satisfying the condition that wH (x) ≡ 0 (mod 8) for all x ∈ C 0 then X k(k−1) τ2 (n, k) · 21+ 2 . NdIV −II (n) = k≤ n 2
Proof. Similar to that of Theorem 5.3, using Proposition 2.3. Note that if the Euclidean weights of all rows of a generator matrix of a self-dual code D are divisible by eight then D is Type II. 2 Remark. We give the values of τ2 (n, k) for small n, which are used in Section 6: • τ2 (8, 0) = 0, τ2 (8, 1) = 1, τ2 (8, 2) = 0, τ2 (8, 3) = 0, τ2 (8, 4) = 0, • τ2 (16, 0) = 0, τ2 (16, 1) = 1, τ2 (16, 2) = 6435, τ2 (16, 3) = 2627625, τ2 (16, 4) = 60810750, τ2 (16, 5) = 64864800, τ2 (16, 6) = 0, τ2 (16, 7) = 0, τ2 (16, 8) = 0.
5.2
Mass Formulas of Type IV F2 + uF2 -Codes
Lemma 5.5 Let C be a self-dual code over F2 + uF2 . Let x and y be codewords of C. Then wH (x + y) ≡ wH (x) + wH (y) (mod 2). Proof. From the definition of the Hamming weight, wH (x + y) = wH (x) + wH (y) − 2(n1,1 + nu,u + n1+u,1+u ) − (n1,1+u + n1,u + nu,1+u ). Since C is self-dual, n1,1+u + n1,u + nu,1+u ≡ 0 Thus wH (x + y) ≡ wH (x) + wH (y)
(mod 2).
(mod 2).
The following corollary is useful when we check if a given code is Type IV.
19
2
Corollary 5.6 Let G be a generator matrix of a self-dual code C over F2 + uF2 . C is Type IV if and only if the Hamming weight of every row in G is even. Corollary 5.6 allows us to give a complete description of Type IV F2 + uF2 -codes in the following theorem. Theorem 5.7 Let C be a code over F2 + uF2 . Suppose that C (1) and C (2) have generator matrices given by (5) and (6), respectively. If C is Type IV, then there exists a unique (1, 0)-matrix B such that I + uB A B k 1 1 (13) 0 uIk2 uC is a generator matrix of C. Moreover, we have ⊥
1) C (2) = C (1) , 2) The residue code C (1) is a self-orthogonal code containing the all-ones vector, 3) The number of u’s in each row of Ik1 + uB is even, and the matrix B is symmetric. Conversely, if C (1) and C (2) are binary codes with generator matrices given by (5) and (6), respectively, and if the conditions 1)–3) are satisfied, then the F2 +uF2 -code C with generator matrix (13) is a Type IV code. Proof. Similar to that of Theorem 5.2, using Section III of [15] and Corollary 5.6.
2
Theorem 5.8 Let Nd0 IV (n) be the number of distinct Type IV F2 + uF2 -codes of length n and let τ3 (n, k) be the number of distinct self-orthogonal codes of length n and dimension k, containing the all-ones vector, then Nd0 IV (n) =
X
τ3 (n, k) · 21+
k(k−1) 2
.
k≤ n 2
2
Proof. Similar to that of Theorem 5.3.
Remark. τ3 (n, k) is the number of self-orthogonal subspaces of dimension k − 1 in a symplectic geometry of dimension n − 2. The formula for τ3 (n, k) is found in Ex. 8.1 of [33] τ3 (n, k) =
k−2 Y i=0
2n−2i−2 − 1 , 2i+1 − 1
where τ3 (n, 0) = 0 and τ3 (n, 1) = 1. codeword 20
Corollary 5.9 Let Nd0 IV −II (n) be the number of distinct Type IV-II F2 + uF2 -codes of length n and let σ1 (n, k) be the number of distinct doubly-even codes of length n and dimension k, containing the all-ones vector, then Nd0 IV −II (n) =
σ1 (n, k) · 21+
X
k(k−1) 2
.
k≤ n 2
Proof. Similar to that of Theorem 5.8, using Proposition 2.5. Note that if the Lee weights of all rows of a generator matrix of a self-dual code D are divisible by four then D is Type II [10]. 2 Remark. σ1 (n, k) is given in [15] and in this case the mass formula is the same as for Type II codes over Z4 .
5.3
Mass Formulas of Type IV F2 + vF2 -Codes
By Corollary 3.4, we have the following: Theorem 5.10 The number of distinct Euclidean Type IV F2 + vF2 -codes of length n is equal to the number of distinct binary self-dual codes of length n. By Corollary 3.6, we have the following: 00 Theorem 5.11 Let NIV (n) be the number of distinct Type IV F2 + vF2 -codes of length n. Then n−1 Y 2n−i−1 − 1 X k−1 00 ). ( NIV (n) = 2i − 1 k=1 i=1
Proof. Let C be a binary code of length n. Both C and its dual are even iff we have the chain of inclusions Rn ⊆ C ⊆ Pn . The all-ones vector spans the 1-dimensional space Rn , and the parity check code Pn is of dimension n − 1. Thus we need to count the number of subspaces of an (n − 1)-dimensional space containing a fixed 1-dimensional subspace. By the homomorphism theorem, such subspaces are in one-to-one correspondence with the subspaces of the quotient space of the (n − 1)-dimensional space by the 1-dimensional space, which is of dimension n − 2. Their number is given in [23, Chap. 24]. 2
21
6
Classification of Type IV Codes of Some Lengths
In this section, we give a complete classification of Type IV codes over Z4 and F2 + uF2 for reasonable lengths. The mass formulas in Section 5 show that our classification is complete.
6.1
Classification of Type IV Codes over Z4
In this subsection, using the classification of self-dual codes over Z4 of length up to 15 given in [9] and [14], and the classification of Type II codes of length 16 in [27], we give a complete classification of Type IV codes for lengths 4, 8 and 12 and we give a complete classification of Type IV-II codes for length 16. We also give the Gray map image φ(C) for each code C. We use the standard notation for binary self-dual codes (see e.g. [30]). As a check, we verified that the mass formula coincides with X C
2n n! , |Aut(C)|
where C runs over all inequivalent codes for each case and |Aut(C)| denotes the order of the automorphism group of C. The values of NdIV (n) and NdIV −II (n) are listed in Tables 1 and 2. Table 1: NdIV (4), NdIV (8) and NdIV (12) Length n 4 NdIV (n) 2
8 142
12 94382
Table 2: NdIV −II (8) and NdIV −II (16) Length n NdIV −II (n)
8 2
16 140626989362
• Length 8 (Type IV-II): K8 is a unique Type IV-II code of length 8. φ(K8 ) is d16 . • Length 16 (Type IV-II): There are five inequivalent Type IV-II codes. These codes are the 5 codes in [27], whose residue codes have no codewords of Hamming weight 4, namely: 1 f 1 (K16 ), 2 f 2 (K82 ), 3 f 3, 4 f 4 and 5 f 5. The Gray map images of them are C1, C5, C29, C67 and C82 in 22
Table A of [7], respectively. Only φ(5 f 5) is an extremal Type II code. Thus 5 f 5 is the only Type IV-II code with dL = 8. • Length 4 (Type IV-I): There is a unique Type IV code, namely D4⊕ (which is equivalent to K4 ). φ(D4⊕ ) is the extended Hamming code e8 . • Length 8 (Type IV-I): 2 2 There is a unique Type IV-I code, namely D4⊕ and φ(D4⊕ ) is 2e8 . • Length 12 (Type IV-I): 3 There are exactly four inequivalent Type IV-I codes, namely D4⊕ , D4⊕ + K8 , K12 and 3 the code [12, 3]-3d4b in [14]. We have verified that φ(D4⊕ ) = 3e8 , φ(D4⊕ +K8 ) = e8 +d16 , φ(K12 ) = d24 and φ([12, 3]-3d4b) = 3d8 .
6.2
Classification of Type IV Codes over F2 + uF2
All Type II codes of lengths 4 and 8 were classified in [10]. Recently a method to extend the classification to larger lengths has been developed in [2]. The classification of self-dual codes over F2 + uF2 is induced from the classification of binary self-dual codes using some property of their automorphism groups [2]. Type I codes of length up to 8 were classified in [10]. By the classification given in [2] and [10], we give the classification of Type IV-II codes of length up to 12 and Lee-optimal Type IV-II codes of length 16 (see Section 7 for the definition of Lee-optimality). For Type IV-I codes, we give the classification of length up to 8 using the classification given in [10], and we give the classification of Lee-optimal Type IV-I codes of length up to 14 using a method in [2]. Similarly to Type IV codes over Z4 , we list the values of Nd0 IV (n) and Nd0 IV −II (n) in Tables 3 and 4 for small lengths n. These values show that the classification is complete for each length and type. Table 3: Nd0 IV (2), Nd0 IV (4), Nd0 IV (6) and Nd0 IV (8) Length n 2 Nd0 IV (n) 2
4 16
6 302
8 22574
• Length 4 (Type IV-II): There is a unique Type IV-II code of length 4, namely K4 .
23
Table 4: Nd0 IV −II (4), Nd0 IV −II (8) and Nd0 IV −II (12) Length n 4 Nd0 IV −II (n) 2
8 5662
12 61711982
• Length 8 (Type IV-II): There are exactly ten Type II codes [10]. By Corollary 5.6, K8 , [8, 2] 2d 4, [8, 3] d 8a and [8, 4] e 8a in [10] are Type IV. The minimum Lee weights dL of the four codes are 4. The minimum Hamming weights dH of [8, 4] e 8a and K8 are 4 and 2, respectively, and the minimum Euclidean weights dE of [8, 4] e 8a and K8 are 4 and 8, respectively. The remaining two codes have dH = 2 and dE = 4. • Length 12 (Type IV-II): Recently the classification of Type II codes was given in [2]. By Corollary 5.6, the fourteen codes A12,6 , A12,8 , B12,1 , C12,1 , C12,4 , C12,8 , C12,20 , E12,1 , E12,6 , F12,3 , EE12,1 , EE12,5 , DE12,1 and DE12,3 in [2] are Type IV. All codes have dL = 4 and dH = 2. Only C12,1 , E12,1 and F12,3 have dE = 8 and the others have dE = 4. The orders of the automorphism groups are 8847360, 294912, 645120, 42467328, 393216, 196608, 73728, 980995276800, 2949120, 491520, 42467328, 516096, 990904320 and 1179648, respectively. • Length 16 (Type IV-II): For this length, we consider only Type IV-II codes with dL = 8 (that is, Lee-optimal). There are a large number of Type II codes of length 16 [2]. By Corollary 5.6, the eight codes C821 , C822 , C831 , C832 , C833 , C841 , C842 and C851 in [2] are Type IV-II codes. All of the codes have dE = 8 and dH = 4. • Lengths 2 and 4 (Type IV-I): K2 is a unique Type IV-I code of length 2 and 2K2 in [10] is a unique Type IV-I code of length 4. • Length 6 (Type IV-I): There are exactly four inequivalent Type IV-I codes of length 6, namely K6 , [6, 3] 3d2 d, [6, 3] 3d2 a and [6, 2] d4 d2 a in [10]. The minimum Lee, Hamming and Eucldiean weights (dL , dH , dE ) of the four codes are (4, 2, 6), (4, 2, 4), (2, 2, 2) and (2, 2, 2), respectively. • Length 8 (Type IV-I): As described above, all Type I codes of length 8 were found, however their generator matrices are not listed in [10]. By a method in [2], we classify the Type IV-I codes and give generator matrices. There are exactly six inequivalent Type IV-I codes of 24
length 8, while two Type IV-I codes with dL = 4 exist. Let HM I8,i be the code with generator matrix G8,i where
1 0 0 0 0 0 0 0 0 1
0 1 0 G8,1 = 0 0 1
0 u 1 u
u 0 u 1
1 u 0 u
u 1 u 0
, G8,2 =
1 0 0 0 0 0 0 0 0 1
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
1 0 0 0 0 0
0 1 0 0 u 0
0 1 0 0 0 u
1 0 0 0 0 0
0 1 u u u u
0 1 0 G8,3 = 0 0 1
G8,5 =
0 1 0 0 0 0
0 1 u 0 0 0
0 1 0 u 0 0
, G8,4 =
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
1 0 0 u 0
1 0 0 0 u
u t u 0 0
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 1 u 0
0 0 1 0 u
0 1 0 0 0
t u u u u 1 0 0 0 0
u u t 0 0 0 0 t u u
, ,
1 0 0 1 , G8,6 = 0 0 0 0
0 0 1 0
0 0 0 1
1 0 0 0 0 u t u . 0 t u u 0 u u t
The orders of their automorphism groups are 211 , 211 .3, 211 .3, 211 .3, 211 .32 .5 and 29 .3, respectively. The minimum Lee, Hamming and Eucldiean weights (dL , dH , dE ) of the six codes HM I8,1 , . . . , HM I8,6 are (4, 2, 4), (4, 2, 4), (2, 2, 2), (2, 2, 2), (2, 2, 2), and (2, 2, 2), respectively. • Length 10 (Type IV-I): There are exactly seven inequivalent binary Type I [20, 10, 4] codes [29]. By a method in [2], these codes construct exactly fourteen Type IV-I codes with dL = 4 of length 10. We give generator matrices G10,i of the codes HM I10,i for i = 1, 2, . . . , 14:
G10,1 =
1 0 0 0 0 0 0 0 0
1 u 0 0 0 0 0 0 0
1 0 u 0 0 0 0 0 0
1 0 0 u 0 0 0 0 0
1 0 0 0 u 0 0 0 0
1 0 0 0 0 u 0 0 0
1 0 0 0 0 0 u 0 0
1 0 0 0 0 0 0 u 0
1 0 0 0 0 0 0 0 u
25
1 u u u u u u u u
1 0 0 1 , G10,2 = 0 0 0 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
u u t u u
u t u u u
u u u t u
t u u u u
u u u u t
,
G10,3 = G10,5 =
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
1 0 u 0 0 0 0 0
1 0 0 u 0 0 0 0
0 1 0 0 u 0 0 0
0 1 0 0 0 u 0 0
0 1 0 0 0 0 u 0
0 1 0 0 0 0 0 u
t 0 u u 0 0 0 0
0 1 0 0 u u u u
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
1 0 0 u 0 0 0
1 0 0 0 u 0 0
1 0 0 0 0 u 0
1 0 0 0 0 0 u
u t u 0 0 0 0
t u u u u u u
u u t 0 0 0 0
1 0 0 0 0 0 1 0 0 0 0
0 0 0 0 1
0 0 1 u u
0 1 0 u u
u u u t u
1 0 0 u u
u u u u t
1 0 0 0 0 0 0
1 0 0 0 u 0 0
0 1 0 0 0 u 0
0 1 0 0 0 0 u
u 1 u 0 0 u u
1 u u u u 0 0
u u 1 0 0 0 0
G10,9 =
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0
0 0 0 0 1
0 0 t u u
u t 0 0 u
0 0 u t u
t u 0 0 u
u u u u t
1 0 0 0 0
0 0 0 1 0
0 0 0 0 1
t t 0 1 0
u 1 1 1 u
1 u 1 1 u
t t 1 0 0
u u 0 0 1
G10,13 =
0 0 1 0 0
26
0 0 0 1 0 0
1 0 0 0 u 0
1 0 0 0 0 u
0 t u u 0 0
0 u t u 0 0
t 0 0 0 u u
0 u u t 0 0
1 0 0 u 0 0 0
1 1 1 u u u u
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 1 0 u 0 0 0
1 1 1 0 u 0 0
1 1 1 0 0 u 0
1 1 1 0 0 0 u
0 0 1 u 0 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 1 1 1 u 0
0 1 1 1 0 u
t 0 1 1 0 0
1 1 t 0 0 0
, G10,12 =
, G10,14 =
t t 0 1 0 0
0 1 t t u u
,
, ,
0 0 1 0 0 0 0 0
0 0 0 1 0 0
1 0 0 0 u 0
1 1 1 0 0 u
u u 1 u u 0
t t 1 u u u
0 1 0 0 u 0
u u 0 t 0 0
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
1 0 0 0 u 0
1 0 0 0 0 u
u 1 0 u 0 0
u 0 1 u 0 0
0 u u t 0 0
t u u 0 u u
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
0 u t u 0
1 0 0 u u
u u 0 0 1
0 1 u 0 u
u 0 u t 0
1 0 0 1 0 0 , G10,10 = 0 0 0 0
1 0 0 0 1 0 0 0 0 0 1 0 0 0
, G10,8 =
0 1 G10,11 = 0 0 0 0
1 0 0 u 0 0 0
, G10,6 =
0 1 0 G10,7 = 0 0 1 0 0 0
1 0 0 0 1 0 0 0 1 , G10,4 = 0 0 0 0 0 0 0 0 0
,
,
,
respectively. The 13 codes HM I10,2 , . . . , HM I10,14 have dL = 4, dH = 2 and dE = 4 while HM I10,1 has dL = 4, dH = 2 and dE = 8. • Lengths 12 and 14 (Type IV-I): We give a classification of Lee-optimal Type IV codes for lengths 12 and 14 (i.e. dL = 6). There is a unique binary self-dual [24, 12, 6] code, namely Z24 in [30]. We checked the automorphism group then three Type I codes are constructed from Z24 by a method in [2], By Corollary 5.6, only one of them is Type IV. We denote this code by HM I12 . A generator matrix G12 of the code is G12 =
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
1 1 1 0 u 0
0 0 1 1 1 u
1 1 1 1 1 0
1 0 1 0 1 1
u t u 1 0 t t t t t t 0 u 0 u 1 u u 1 u t 1 t u
.
Its minimum Hamming and Euclidean weights are 4 and 6, respectively. There are exactly three inequivalent binary self-dual [28, 14, 6] codes A28 , B28 and C28 in [6] (see also [7]). The numbers of Type I codes over F2 + uF2 constructed from the three binary codes are 1, 4 and 3, respectively. Only one of the eight Type I codes is a Type IV code with dL = 6 denoted by HM I14 . The code is obtained from B28 . This code has dH = 4 and dE = 6 and G14 is its generator matrix. G14 =
7
1 0 0 0 0 0 0
0 1 0 0 0 0 0
0 0 1 0 0 0 0
0 0 0 1 0 0 0
1 1 1 0 u 0 0
0 1 1 1 0 u 0
1 1 0 1 0 0 u
0 1 0 1 1 0 0
1 1 0 1 0 1 0
1 0 0 1 1 0 1
u 0 1 1 t t u
t t 1 t t t t
u 0 0 t 0 1 1
0 1 t 1 u u t
.
Optimal Type IV Codes
We determine the highest possible minimum weight of codes by examining the possible symmetrized weight enumerators of the code and its shadow using the rings for the symmetrized weight enumerators given in Section 4 noting the coefficients must be non-negative integers for each class. As an example, we consider Type IV codes over F2 + uF2 of length 6. We assume that the highest possible Hamming weight is 4. Any code with minimum Hamming weight 4 has the following symmetrized weight enumerator: W6 = 15a2 b4 + 15b4 c2 + 30a2 b2 c2 + a6 + c6 + 2b6 27
and its shadow has symmetrized weight enumerator: S6 =
15 4 2 15 2 4 a b + b c + ···. 2 2
W6 has non-negative integral coefficients but S6 has non-integral coefficients. Thus the highest minimum Hamming weight is 2. We say that a Type IV-I (resp. Type IV-II) code C is Lee-optimal, Euclidean-optimal and Hamming-optimal if C has highest minimum Lee, Euclidean and Hamming weight among all Type IV-I (resp. Type IV-II) codes of that length, respectively. In this section, dL (n), dH (n) and dE (n) denote the highest minimum Lee, Hamming and Euclidean weights of length n, respectively for each class.
7.1
Type IV Codes over Z4
In Tables 5 and 6, we list the highest minimum weights for Type IV-I and Type IV-II codes over Z4 of length up to 24, together with examples of optimal Type IV codes. All optimal codes given in the tables were given in previous sections. At lengths where the highest minimum weight is not determined, we construct some codes with high minimum weight. Several such codes are also constructed from optimal codes of small lengths by the direct sum. For example, a Type IV-I code with dE = 8 of length 24 is constructed from the direct sum of K12 and K12 . However it is not known if there is a Type IV-I code with dE = 12. Table 5: The Highest Minimum Weights for Type IV-I Z4 -Codes Length n 4 8 12 16 20 24
dL (n) 4 4 4 4 4 4 or 8
Codes D4⊕ 2 D4⊕ the four codes all codes all codes ?
dH (n) 2 2 2 2 or 4 2 2 or 4
Codes D4⊕ 2 D4⊕ the four codes ? all codes ?
dE (n) 4 4 8 4 or 8 8 8 or 12
Codes D4⊕ 2 D4⊕ K12 , [12, 3]-3d4b ? K20 ?
Table 6: The Highest Minimum Weights for Type IV-II Z4 -Codes Length n 8 16 24
dL (n) 4 8 4
Codes K8 5f 5 any code
dH (n) 2 4 2
28
Codes K8 5f 5 any code
dE (n) 8 8 8
Codes K8 the five codes any code
7.2
Type IV Codes over F2 + uF2
In Tables 7 and 8, we list the highest minimum weights for Type IV-I and Type IV-II codes over F2 + uF2 of length up to 24, together with examples of optimal Type IV codes. All examples are given here or in Section 6. Table 7: The Highest Minimum Weights for Type IV-I F2 + uF2 -Codes Length n 2 4 6 8 10 12 14 16 18 20 22 24
dL (n) 2 2 4 4 4 6 6 8 6 or 8 8 8 10
Codes all codes all codes K6 , [6, 3] 3d2 d HM I8,1 , HM I8,2 HM I10,1 , . . . , HM I10,14 HM I12 HM I14 HM I16 (dL (18) = 6 [16]) HM I20 HM I22 HM I24
dH (n) 2 2 2 2 2 4 4 4 4 4 or 6 6 6 or 8
Codes all codes all codes all codes all codes all codes HM I12 HM I14 HM I16 [16] ? HM I22 ?
dE (n) 2 2 6 6 8 8 8 8 8 8, 10 or 12 8, 10 or 12 10 or 12
Codes all codes all codes K6 HM I8,1 , HM I8,2 HM I10,1 [16] [16] HM I16 [16] ? ? ?
Table 8: The Highest Minimum Weights for Type IV-II F2 + uF2 -Codes Length n 4 8 12 16 20 24
dL (n) 4 4 4 8 8 8 or 12
Codes all codes all codes all codes the eight codes HM20 (dL (24) = 8 [16])
dH (n) 2 4 2 4 4 or 6 8
Codes all codes [8, 4] e 8a all codes the eight codes ? [16]
dE (n) 4 8 8 8 8 or 12 8 or 12
Codes all codes K8 C12,1 , E12,1 and F12,3 the eight codes ? ?
We construct optimal codes listed in Tables 7 and 8. The codes are obtained from binary self-dual codes by a method in [2]. Let HM20 be the Type IV-II code with generator matrix G20 where
29
G20
=
1 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 1 0 0
1 1 t 0 1 1 t 0 0 1 0 1 0 1 0 u 1 1 1 1 u 1 1 1 0 1 t 1 u u 1 0 t 1 0 1 t t u u t t 1 0 1 1 0 1 0 1 0 0 t 1 1 1 0 0 t 1 u 1 1 1 1 u , 1 1 u u u 0 0 1 0 0 u 1 1 t u 1 0 1 u 0 t 1 1 0 1 t 0 1 t t u u t u 0 0 0 u u 0 u 0 u u 0 u u u u u u u u u u
where t denotes 1 + u in the matrices. HM20 is a code with dL = 8, dH = 4 and dE = 8, thus the code is Lee-optimal but not Euclidean-optimal nor Hamming-optimal. Generator matrices of Type IV-I codes HM I16 , HM I20 , HM I22 and HM I24 listed in Table 7 are the following:
G16
=
G20
=
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0 0
0 0 1 0 1 0 0 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 u 0 0 0 u
u t t 1 1 u 0 t u t 0 t t u 1 1 t t u u 0 u u 1 u u u u 1 u t 0 1 u 1 , 0 1 u 0 u t 1 u 0 u 1 0 1 t u u u u u 0 0 0 u u u 0 u u
1 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0
0 t 0 t t 0 u 0 t t u 0 u 1 0 t t 0 u 0 t t 0 1 u t 0 t t 0 u 0 t 0 1 t u t 0 t t 0 u 0 0 0 t 1 u t 0 t t 0 u , 0 u 0 1 1 u t 0 t t 0 0 0 u 0 1 1 u t 0 t t 0 1 0 u 0 1 1 u t 0 t 0 1 t 0 u 0 1 1 u t 0 1 0 t 1 0 u 0 1 1 u t
0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 1 0
30
G22
=
G24
=
1 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 1
t 0 t 0 0 t 0 u u t t 1 t 0 t 0 0 t 0 u u t 1 1 t 0 t 0 0 t 0 u u u 1 1 t 0 t 0 0 t 0 u u u 1 1 t 0 t 0 0 t 0 0 u u 1 1 t 0 t 0 0 t , 1 0 u u 1 1 t 0 t 0 0 0 1 0 u u 1 1 t 0 t 0 0 0 1 0 u u 1 1 t 0 t 1 0 0 1 0 u u 1 1 t 0 0 1 0 0 1 0 u u 1 1 t
1 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0 0 0 1
t 0 0 t 0 0 0 u t u t t 1 t 0 0 t 0 0 0 u t u t 1 1 t 0 0 t 0 0 0 u t u u 1 1 t 0 0 t 0 0 0 u t 1 u 1 1 t 0 0 1 0 0 0 u u 1 u 1 1 t 0 0 t 0 0 0 , 0 u 1 u 1 1 t 0 0 t 0 0 0 0 u 1 u 1 1 1 0 0 t 0 0 0 0 u 1 u 1 t t 0 0 t t 0 0 0 u t u 1 t 1 0 0 0 1 0 0 0 u 1 u 1 1 t 0 0 0 1 0 0 0 u t u 1 1 t
respectively. HM I16 is a Lee, Euclidean and Hamming-optimal Type IV-I code of length 16. HM I20 is a Lee-optimal Type IV-I code with dH = 4 and dE = 8 of length 20. HM I22 is a Lee-optimal Type IV-I code with dH = 6 and dE = 8 of length 22. HM I24 is a Lee-optimal Type IV-I code with dH = 6 and dE = 10 of length 24.
7.3
Type IV Codes over F2 + vF2
As described in Section 3, the classification of binary self-dual codes leads to the classification of Euclidean Type IV codes over F2 + vF2 . Since all the binary self-dual codes of length up to 30 are classified, one can determine the highest minimum weights of Euclidean Type IV codes over F2 + vF2 of length up to 30. Thus here we consider only Hermitian Type IV codes. For length up to 24, we list in Table 9 only the highest possible minimum weights for Type IV codes over F2 + vF2 . In the table, dB (n), dL (n) and dH (n) denote the highest possible minimum Bachoc, Lee and Hamming weights of length n, respectively. For length n up to 18, we also give the possible symmetrized weight enumerators WB (n) with highest minimum Bachoc weight. WB (2) = a2 + 2b2 + c2 WB (4) = a4 + 6a2 b2 + 2b4 + 6b2 c2 + c4
31
WB (6) = a6 + 15a2 b4 + 30a2 b2 c2 + 2b6 + 15b4 c2 + c6 WB (8) = 28 a4 b2 c2 + 14 a4 b4 + 28 a2 b6 + 14 b4 c4 + 28 b6 c2 + a8 + 2 b8 + c8 + 112 a2 c2 b4 +28 a2 b2 c4 WB (10) = 90 a4 b2 c4 + 360 a2 b6 c2 + 180 a2 b4 c4 + 180 a4 b4 c2 + 30 a4 b6 + 45 a2 b8 + 45 b8 c2 +30 b6 c4 + 30 b4 c6 + 30 a6 b4 + a10 + 2 b10 + c10 WB (12) = c12 + (81 α + 9) c8 b4 + 66 c6 a4 b2 + (−324 α + 228) c6 a2 b4 + 66 c6 b6 +66 c4 a6 b2 + (486 α + 516) c4 a4 b4 + 858 c4 a2 b6 + 99 c4 b8 + (−324 α + 228) c2 a6 b4 + 858 c2 a4 b6 + 792 c2 a2 b8 + 66 c2 b10 + a12 + (81 α + 9) a8 b4 + 66 a6 b6 + 99 a4 b8 + 66 a2 b10 + 2 b12 WB (14) = 1274 a6 b6 c2 + 182 a6 b2 c6 + 273 a8 b4 c2 + 728 a6 b4 c4 + 2730 a4 b8 c2 + 728 a4 b4 c6 +3458 a4 b6 c4 + 1638 a2 b10 c2 + 1274 a2 b6 c6 + 2730 a2 b8 c4 + 273 a2 b4 c8 + a14 +2 b14 + c14 + 182 b10 a4 + 182 b10 c4 + 273 a6 b8 + 91 a2 b12 + 91 b12 c2 + 273 b8 c6 WB (16) = c16 + (324 α − 240) c10 b6 + (−324 α + 408) c10 b4 a2 + (−324 α + 438) c8 b8 + (−972 α + 2184) c8 b6 a2 + (1296 α − 132) c8 b4 a4 + 120 c8 b2 a6 + 568 c6 b10 + (1296 α + 4608) c6 b8 a2 + (648 α + 6064) c6 b6 a4 + (−1944 α + 3088) c6 b4 a6 +120 c6 b2 a8 + 380 c4 b12 + 7440 c4 b10 a2 + (−1944 α + 15648) c4 b8 a4 + (648 α + 6064) c4 b6 a6 + (1296 α − 132) c4 b4 a8 + 120 c2 b14 +2880 c2 b12 a2 + 7440 c2 b10 a4 + (1296 α + 4608) c2 b8 a6 + (−972 α + 2184) c2 b6 a8 + (−324 α + 408) c2 b4 a10 + 2 b16 + 120 b14 a2 +380 b12 a4 + 568 b10 a6 + (−324 α + 438) b8 a8 + (324 α − 240) b6 a10 + a16 WB (18) = c18 + (−729 α + 108) c12 b6 + 1224 c10 a4 b4 + (4374 α + 576) c10 a2 b6 +306 c10 b8 + 306 c8 a8 b2 + 1836 c8 a6 b4 + (−10935 α + 9576) c8 a4 b6 +7956 c8 a2 b8 + 306 c8 b10 + 1836 c6 a8 b4 + (14580 α + 16608) c6 a6 b6 +35496 c6 a4 b8 + 19584 c6 a2 b10 + 1428 c6 b12 + 1224 c4 a10 b4 + (−10935 α + 9576) c4 a8 b6 + 35496 c4 a6 b8 + 47736 c4 a4 b10 +17136 c4 a2 b12 + 612 c4 b14 + (4374 α + 576) c2 a10 b6 + 7956 c2 a8 b8 +19584 c2 a6 b10 + 17136 c2 a4 b12 + 4896 c2 a2 b14 + 153 c2 b16 + a18 + (−729 α + 108) a12 b6 + 306 a10 b8 + 306 a8 b10 + 1428 a6 b12 + 612 a4 b14 +153 a2 b16 + 2 b18 ,
where α is a parameter. It is a worthwhile project for someone to determine the exact highest minimum weights by constructing optimal codes.
32
Table 9: The Highest Possible Minimum Weights for Type IV F2 + vF2 -Codes Length n 2 4 6 8 10 12
dB (n) 2 4 6 6 8 8
dL (n) 2 4 4 4 6 6
dH (n) 2 2 4 4 4 6
Length n 14 16 18 20 22 24
dB (n) 10 10 12 12 14 14
dL (n) 6 8 10 10 12 12
dH (n) 6 6 8 8 8 10
References [1] C. Bachoc, “Application of coding theory to the construction of modular lattices,” J. Combin. Theory Ser. A, vol. 78, pp. 92–119, 1997. [2] E. Bannai, M. Harada, A. Munemasa and M. Oura, “Type II codes over F2 + uF2 and applications to hermitian modular forms,” (in preparation). [3] A. Bonnecaze, P. Sol´e and A.R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices,” IEEE Trans. Inform. Theory, vol. IT-41, pp. 366–377, 1995. [4] A. Bonnecaze, P. Sol´e, C. Bachoc and B. Mourrain, “Type II codes over Z4 ,” IEEE Trans. Inform. Theory, vol. IT-43, pp. 969–976, 1997. [5] A.R. Calderbank and N.J.A. Sloane, “Double circulant codes over Z4 and even unimodular lattices,” J. Alg. Combin., vol. 6, pp. 119–131, 1997. [6] J.H. Conway and V. Pless, “On the enumeration of self-dual codes,” J. Combin. Theory Ser. A, vol. 28, pp. 26–53, 1980. [7] J.H. Conway, V. Pless and N.J.A. Sloane, “The binary self-dual codes of length up to 32: a revised enumeration,” J. Combin. Theory Ser. A, vol. 60, pp. 183–195, 1992. [8] J.H. Conway and N.J.A. Sloane, Sphere Packing, Lattices and Groups (2nd ed.), New York: Springer-Verlag, 1993. [9] J.H. Conway and N.J.A. Sloane, “Self-dual codes over the integers modulo 4,” J. Combin. Theory Ser. A, vol. 62, pp. 30–45, 1993. [10] S.T. Dougherty, P. Gaborit, M. Harada, and P. Sol´e, “Type II codes over F2 + uF2 ,” IEEE Trans. Inform. Theory, (to appear). [11] S.T. Dougherty, M. Harada and P. Sol´e, “Self-dual codes over rings and the Chinese remainder theorem,” Hokkaido Math. J., (to appear). 33
[12] S.T. Dougherty, M. Harada and P. Sol´e, “Shadow codes over Z4 ,” (submitted). [13] S.T. Dougherty, P. Gaborit, and P. Sol´e, “Self-dual codes over F2 + vF2 ,” (in preparation). [14] J. Fields, P. Gaborit, J. Leon and V. Pless, “All self-dual Z4 codes of length 15 or less are known,” IEEE Trans. Inform. Theory, vol. IT-44, pp. 311–322, 1998. [15] P. Gaborit, “Mass formula for self-dual codes over Z4 and Fq + uFq rings,” IEEE Trans. Inform. Theory, vol. IT-42, pp. 1222–1228, 1996. [16] T.A. Gulliver, Private communication, September 1998. [17] T.A. Gulliver and M. Harada, “Extremal double circulant Type II codes over Z4 and construction of 5-(24, 10, 36) designs,” Discrete Math., (to appear). [18] A.R. Hammons, Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane and P. Sol´e, “The Z4 -linearity of Kerdock, Preparata, Goethals and related codes,” IEEE Trans. Inform. Theory, vol. IT-40, pp. 301–319, 1994. [19] M. Harada, “New 5-designs constructed from the lifted Golay code over Z4 ,” J. Combin. Designs, vol. 6, pp. 225–229, 1998. [20] M. Harada, P. Sol´e and P. Gaborit, “Self-dual codes over Z4 and unimodular lattices: a survey,” (submitted). [21] M. Klemm, “Selbstduale Codes u ¨ber dem Ring der ganzen Zahlen modulo 4,” Archiv. Math., vol. 53, pp. 201–207, 1989. [22] C.W.H. Lam and V. Pless, “There is no (24, 12, 10) self-dual quaternary code,” IEEE Trans. Inform. Theory, vol. IT-36, pp. 1153–1156, 1990. [23] J.H. van Lint and R.M. Wilson, A Course in Combinatrics, Cambridge: Cambridge University Press, 1996. [24] F.J. MacWilliams, C.L. Mallows and N.J.A. Sloane, “Generalizations of Gleason’s theorem on weight enumerators of self-dual codes,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 794–805, 1972. [25] F.J. MacWilliams, A.M. Odlyzko, N.J.A. Sloane and H.N. Ward, “Self-dual codes over GF (4),” J. Combin. Theory Ser. A, vol. 25, pp. 288–318, 1978. [26] C.L. Mallows and N.J.A. Sloane, “An upper bound for self-dual codes,” Inform. Control, vol. 22, pp. 188–200, 1973.
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[27] V. Pless, J. Leon and J. Fields, “All Z4 codes of Type II and length 16 are known,” J. Combin. Theory Ser. A, vol. 78, pp. 32–50, 1997. [28] V. Pless, “The number of isotropic subspaces in a finite geometry,” Atti. Accad. Naz. Lincei Rendic, vol. 39, pp. 418–421, 1965. [29] V. Pless, “A classification of self-orthogonal codes over GF (2),” Discrete Math., vol. 3, pp. 209–246, 1972. [30] V. Pless and N.J.A. Sloane, “On the classification and enumeration of self-dual codes,” J. Combin. Theory Ser. A, vol. 18, pp. 313–335, 1975. [31] E.M. Rains, “Shadow bounds for self-dual codes,” IEEE Trans. Inform. Theory, vol. IT-44, pp. 134–139, 1998. [32] N.J.A. Sloane, “Self-dual codes and lattices,” Proc. Symposia in Pure Math., Amer. Math. Soc., Provid. RI, vol. 34, pp. 273–308, 1979. [33] D. E. Taylor, The Geometry of the Classical Groups, Berlin: Heldermann Verlag, 1992. [34] J. Wood, “Duality for modules over finite rings and applications to coding theory,” (submitted).
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