Lifted Codes over Finite Chain Rings Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510 USA Email: [email protected] Hongwei Liu Department of Mathematics Huazhong Normal University Wuhan, Hubei 430079 P. R. China Email: h w [email protected] Young Ho Park Department of Mathematics Kangwon National University Chuncheon 200-701 Korea Email: [email protected] June 22, 2011

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Abstract In this paper, we study lifted codes over finite chain rings. We define γ-adic codes over a formal power series ring and use these to study codes over finite chain rings. We use a generalized Chinese Remainder Theorem to extend the results to codes over principal ideal rings.

Keywords: Finite chain rings, lifted codes, γ-adic codes. 2000 Mathematical Subject Classification: Primary: 94B05, Secondary: 13A99

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1

Introduction

Codes over finite rings have been studied for many years. More recently, interest in codes over rings has grown rapidly, especially following the landmark paper [7]. This paper, written by Hammons et al., showed that there exists an interesting connection between non-linear binary codes and linear codes over Z4 . In [7], it is proved that some non-linear codes, such as the Kerdock, Preparata, and Goethal codes can be viewed as linear codes over Z4 via the 2 Gray map from Zn4 to F2n 2 , where Z4 is given the Lee metric and F2 the Hamming metric. Following this a large series of papers studying codes over rings have appeared. In [1], Calderbank and Sloane studied codes over the p-adic integers and also gave a description of the lifts of codes over Zp to Zpe and to the p-adics. Later, in [2], Dougherty, Kim and Park studied these codes further and also found the weight enumerators of this class of codes. An important class of rings over which codes have been studied is the class of chain rings, especially Zpe , which are a special case of chain rings. In this paper, we shall first define a series of chain rings and introduce the concept of γ-adic codes. Then we will study these γ-adic codes over this class of chain rings. We begin with some definitions. Throughout we let R be a finite commutative ring with identity 1 6= 0. Let Rn = {(x1 , · · · , xn ) | xj ∈ R} be an R-module. An R-submodule C of Rn is called a linear code of length n over R. We assume throughout that all codes are linear. For x, y ∈ Rn , the inner product of x, y is defined as follows: [x, y] = x1 y1 + · · · + xn yn . If C is a code of length n over R, we define C ⊥ = {x ∈ Rn | [x, c] = 0, ∀ c ∈ C} to be the orthogonal code of C. Notice that C ⊥ is linear whether or not C is linear. In [15], it is proved that for any linear code C over a finite Frobenius ring, |C| · |C ⊥ | = Rn .

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A finite chain ring is a Frobenius ring, so the identity above holds for codes over finite chain rings. If C ⊆ C ⊥ , then C is called self-orthogonal. Moreover, if C = C ⊥ , then C is called self-dual.

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Notations and Finite Chain Rings

A finite ring is called a chain ring if its ideals are linearly ordered by inclusion. In particular, this means that any finite chain ring has a unique maximal ideal. 3

Let R be a finite chain ring, m the unique maximal ideal of R, and let γ be the generator of the unique maximal ideal m. Then m = hγi = Rγ, where Rγ = hγi = {βγ | β ∈ R}. We have R = hγ 0 i ⊇ hγ 1 i ⊇ · · · ⊇ hγ i i ⊇ · · · . (2) The chain in (2) can not be infinite, since R is finite. Therefore, there exists an i such that hγ i i = {0}. Let e be the minimal number such that hγ e i = {0}. The number e is called the nilpotency index of γ. Let |R| denote the cardinality of R and R× the multiplicative group of all units in R. Let F = R/m = R/hγi be the residue field with characteristic p, where p is a prime number. We know that |F| = q = pr for some integers q and r and |F× | = pr − 1. The following two lemmas are well-known (see [12], for example). Lemma 2.1. Assume the notations given above. For any 0 = 6 r ∈ R there is a unique i integer i, 0 ≤ i < e such that r = µγ , with µ a unit. The unit µ is unique modulo γ e−i . Lemma 2.2. Let R be a finite chain ring with maximal ideal m = hγi, where γ is a generator of m with nilpotency index e. Let V ⊆ R be a set of representatives for the equivalence classes of R under congruence modulo γ. Then P i (i) for all r ∈ R there exist unique r0 , · · · , re−1 ∈ V such that r = e−1 i=0 ri γ ; (ii) |V | = |F|; (iii) |hγ j i| = |F|e−j for 0 ≤ j ≤ e − 1. By Lemma 2.2, we can compute the cardinality of R as follows: |R| = |F| · |hγi| = |F| · |F|e−1 = |F|e = per .

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Let R be a finite ring. We know from [12] that the generator matrix for a code C over R is permutation equivalent to a matrix of the following form:   Ik0 A0,1 A0,2 A0,3 A0,e     γIk1 γA1,2 γA1,3 γA1,e   2 2 2   γ I γ A γ A k 2,3 2,e 2   G= (4) . .. .. . .     .. ..   . .   e−1 e−1 γ Ike−1 γ Ae−1,e The matrix G above is called the standard generator matrix form of the code C. It is immediate that a code C with this generator matrix has cardinality Pe−1

|C| = |F|

i=0 (e−i)ki

Pe−1

= (pr )

i=0 (e−i)ki

= (pre )k0 (pr(e−1) )k1 · · · (pr )ke−1 .

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In this case, the code C is said to have type 1k0 (γ)k1 (γ 2 )k2 · · · (γ e−1 )ke−1 . 4

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3

Lifts of Codes over Finite Chain Rings

In this section, we shall first define a series of rings from a fixed finite chain ring. We will discuss some properties about these rings. We then study lifts of codes over finite chain rings. ˜ be a finite chain ring with the maximal ideal h˜ Let R γ i, where the nilpotency index of γ˜ ˜ ˜ it can be written uniquely as is e and R/h˜ γ i = F. We know that for any element a of R, a = a0 + a1 γ˜ + · · · + ae−1 γ˜ e−1 , where ai ∈ F. For an arbitrary positive integer i, we define Ri as Ri = {a0 + a1 γ + · · · + ai−1 γ i−1 | ai ∈ F} where γ i−1 6= 0, but γ i = 0 in Ri , and define two operations over Ri : i−1 X l=0 i−1 X l=0

l

al γ + al γ l ·

i−1 X

l=0 i−1 X

bl γ

bl 0 γ l

l

0

i−1 X = (al + bl )γ l

=

l0 =0

l=0 i−1 X

(

X

al b0l )γ s .

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s=0 l+l0 =s

It is easy to get that all the Ri are finite rings. Moreover, we have the following lemma. Lemma 3.1. Assume the notations given above. Then for any positive integer i, we have i−1 P (i) Ri× = { al γ l | 0 6= a0 ∈ F}; l=0

(ii) the ring Ri is a chain ring with maximal ideal hγi. Proof.

(i) Let a =

i−1 P

al γ l ∈ Ri such that a0 6= 0. We need to show there exists an

l=0

element b =

i−1 P

l0

bl0 γ ∈ Ri such that ab = 1. This means that we need to show the following

l0 =0

equation system a0 b0 = 1, a0 b1 + a1 b0 = 0, · · · , a0 bi−1 + a1 bi−2 + · · · + ai−1 b0 = 0 has solutions for b0 , b1 , · · · , bi−1 in Ri . Since a0 6= 0, there exists a solution b0 to a0 b0 = 1. For s < i assume there are solutions for b0 , b1 , · · · , bs−1 of the first s equations a0 b0 = 1, a0 b1 + a1 b0 = 0, · · · , a0 bs−1 + a1 bs−2 + · · · + as−1 b0 = 0. Then a0 bs + a1 bs−1 + · · · + as b0 = 0 has a solution for bs since a0 6= 0. Then by induction there are solutions for all bl0 . Hence there exists b with ab = 1 and a is a unit. 5

(ii) If i = 1 then R1 = F, and the result follows directly. Now suppose i > 1, let a = a0 + a1 γ + · · · + ai−1 γ i−1 ∈ Ri . We define ρ to be a map from Ri to F given by ρ : Ri → F, a 7→ a0 . It is easy to obtain that the map ρ is a surjective homomorphism. We can get that Ker(ρ) = {a0 + a1 γ + · · · + ai−1 γ i−1 ∈ Ri | ρ(a) = 0} = hγi. This gives that Ri /hγi ∼ = F, and so hγi is a maximal ideal of Ri . Let I be an arbitrary ideal of Ri . If I = {0} then I = h0i. Now suppose I 6= {0}, and I 6= hγ j i for all 1 ≤ j ≤ i − 1, then there exists a ∈ I such that a = a0 + a1 γ + · · · + ai−1 γ i−1 with a0 6= 0. By (i), this implies that a is a unit in Ri . Therefore b = (ba−1 )a ∈ I for any b ∈ Ri , and this implies that Ri = I. Hence Ri is a chain ring. ˜ since there is an isomorphism Remark 1. We note that if i = e then Re is isomorphic to R, as follows: ˜ Re → R, a0 + a1 γ + · · · + ae−1 γ e−1 7→ a0 + a1 γ˜ + · · · + ae−1 γ˜ e−1 . ˜ This allow us sometimes to identify Re with R. We define R∞ as the ring of formal power series as follows: R∞ = F[[γ]] = {

∞ X

al γ l | al ∈ F}.

l=0

We know that for any two elements a =

∞ P

al γ l , b =

∞ P

bl γ l ∈ R∞ , their sum and product

l=0

l=0

bl γ l =

∞ X (al + bl )γ l

are the following: ∞ X

al γ l +

l=0 ∞ X l=0

∞ X l=0

al γ l ·

∞ X

bl 0 γ l

l=0 0

∞ X X = ( al bl0 )γ s .

l0 =0

s=0 l+l0 =s

Then the following lemma is well-known. Lemma 3.2. Assume the notations given above. Then we have that ∞ P × (i) R∞ = { al γ l | a0 6= 0}; l=0

(ii) the ring R∞ is a principal ideal domain. 6

Lemma 3.3. Let C be a nonzero linear code over R∞ of length n, then any generator matrix of C is permutation equivalent to a matrix of the following form:   γ m0 A0,r γ m0 Ik0 γ m0 A0,1 γ m0 A0,2 γ m0 A0,3    γ m1 Ik1 γ m1 A1,2 γ m1 A1,3 γ m1 A1,r     γ m2 A2,r  γ m2 Ik2 γ m2 A2,3   (9) G= , ... ...     ... ...     γ mr−1 Ikr−1 γ mr−1 Ar−1,r where 0 ≤ m0 < m1 < · · · < mr−1 for some integer r. The column blocks have sizes k0 , k1 , · · · , kr and the ki are nonnegative integers adding to n. Proof. Before proving the lemma, we note that all nonzero elements in R∞ can be written in the form γ i a, where a = a0 + a1 γ + · · · + · · · with a0 6= 0 and i ≥ 0. This means that a is a unit in R∞ . Let Ω be an arbitrary set of generators of code C, a generator matrix G can be obtained by eliminating those elements which can be written as a linear combination of other elements in the set Ω. In order to obtain the standard form in this lemma, we do the following operations. First we take one nonzero element with form γ m0 a, where m0 is the minimal nonnegative integer such that m0 = min{i | γ i a is a coordinate in an element of Ω}. By applying column and row permutations and by dividing a row by a unit, the element in position (1, 1) of matrix G can be replaced by γ m0 . Since those nonzero elements which are in the first column of matrix G have the form γ j b with j ≥ m0 and b a unit, these elements can be replaced by zero when they are added by the first row which multiplied by −γ j−m0 b−1 . Then we continue this process by using elementary operations, and the standard form of G is obtained. Remark 2. A code C with generator matrix of the form given in Equation (9) is said to be of type (γ m0 )k0 (γ m1 )k1 · · · (γ mr−1 )kr−1 , where k = k0 + k1 + · · · + kr−1 is called its rank and kr = n − k. A code C of length n with rank k over R∞ is called a γ-adic [n, k] code. We call k the dimension of C and denote the dimension by dim C = k. Lemma 3.4. If C is a linear code over R∞ then C ⊥ has type 1m for some m. Proof. By Equation (9), we know that all codewords in C ⊥ are of the form γ l v for some nonnegative integer l. This implies that [γ l v, w] = 0 for all w ∈ C. Hence we have 0 = [γ l v, w] = γ l

n X l=1

7

vl wl = γ l [v, w].

This gives that [v, w] = 0 since 0 6= γ l ∈ R∞ and R∞ is a domain. This implies that v ∈ C ⊥ . Therefore the code C ⊥ has the type 1m for some m. We denote the transpose of a matrix M by M T . Theorem 3.5. Let C be a linear code of length n over R∞ . If C has a standard generator matrix G as in equation (9), then we have (i) the dual code C ⊥ of C has a generator matrix   H = B0,r B0,r−1 · · · B0,2 B0,1 Ikr , (10) where B0,j = −

j−1 P

B0,l ATr−j,r−l − ATr−j,r for all 1 ≤ j ≤ r;

l=1

(ii) rank(C) + rank(C ⊥ ) = n. Proof. By Lemma 3.4, we know that C ⊥ has type 1m for some m. This gives that C ⊥ has the following generator matrix   H = B0,r B0,r−1 · · · B0,2 B0,1 B0,0 , (11) where B0,j = Im for some j. Since HGT = 0, this implies that for any 0 ≤ j ≤ r we have that B0,r−j Ikj + B0,r−j−1 ATj,j+1 + B0,r−j−2 ATj,j+2 + · · · + B0,1 ATj,r−1 + B0,0 ATj,r = 0. By replacing j with r − j, we get that B0,j = −B0,j−1 ATr−j,r−j+1 − B0,j−2 ATr−j,r−j+2 − · · · − B0,1 ATr−j,r−1 − B0,0 ATr−j,r . Then the result follows by taking B0,0 = Ikr . (ii) By the proof of (i), we know that rank(C ⊥ ) = kr = n − (k0 + k1 + · · · + kr−1 ) = n − rank(C). This implies that rank(C) + rank(C ⊥ ) = n. Example 1. Let C be a code of length 5 over R∞ with a standard generator matrix as follows:   γ 2 0 γ 2 (1 + γ) γ 2 (1 + γ + γ 2 ) γ2   G =  0 γ 2 γ 2 (1 + 2γ) (12) γ 2 (1 + γ 2 ) γ 2 (1 + 3γ 2 )  . 0 0 γ4 γ 4 (1 + γ 2 ) γ 4 (2 + γ) Then the dual code C ⊥ of C has a generator matrix H=

γ3 2γ + 2γ 3 −(1 + γ 2 ) 1 0 1 + 3γ + γ 2 1 + 5γ − γ 2 −(2 + γ) 0 1

This gives that rank(C) + rank(C ⊥ ) = 3 + 2 = 5(= n). 8

! .

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For two positive integers i < j, we define a map as follows: Ψji : Rj → Ri , j−1 i−1 X X al γ l 7→ al γ l . l=0

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l=0

If we replace Rj with R∞ then we denote Ψ∞ i by Ψi . Let a, b be two arbitrary elements in Rj . It is easy to get that Ψji (a + b) = Ψji (a) + Ψji (b), Ψji (ab) = Ψji (a)Ψji (b).

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If a, b ∈ R∞ . We have that Ψi (a + b) = Ψi (a) + Ψi (b), Ψi (ab) = Ψi (a)Ψi (b).

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n to Rin and We note that the two maps Ψi and Ψji can be extended naturally from R∞ Rjn to Rin respectively.

˜ the construction Remark 3. Assume the notations given above. For any fixed chain ring R, method above gives a series of chain rings (up to the principal ideal domain R∞ ) as follows:

R∞ → · · ·

˜ R F k k → Re → Re−1 → · · · → R1

Definition 1. Let i, j be two integers such that 1 ≤ i ≤ j < ∞. We say that an [n, k] code C1 over Ri lifts to an [n, k] code C2 over Rj , denoted by C1  C2 , if C2 has a generator matrix G2 such that Ψji (G2 ) is a generator matrix of C1 . We also denote C1 by Ψji (C2 ). If C is a [n, k] γ-adic code, then for any i < ∞, we call Ψi (C) a projection of C. We denote Ψi (C) by C i . Lemma 3.6. Let M be a matrix over R∞ with type 1k . If M 0 is a standard form of M , then for any positive integer i, Ψi (M 0 ) is a standard form of Ψi (M ). Proof. We note that M has type 1k , hence Ψi (M ) has type 1k . We know M 0 is a standard form of M , this implies that there exist elementary matrices P1 , · · · , Ps and Q1 , · · · , Qt such that P1 · · · Ps M Q 1 · · · Q t = M 0 . Hence for any positive integer i, by Equation (17), we have that Ψi (P1 ) · · · Ψi (Ps )Ψi (M )Ψi (Q1 ) · · · Ψi (Qt ) = Ψi (M 0 ). Since the inverse matrices of elementary matrices are the same type of elementary matrices, we have that Ψi (M 0 ) is a standard form of Ψi (M ). 9

Remark 4. In the lemma above we must assume that M has type 1k . For example, if we take ! γ5 γ5 + γ7 M= , (18) 0 γ 15 then some of its projections are the zero matrix. Let C be a code over R∞ , we know that C ⊆ (C ⊥ )⊥ . But in general C = 6 (C ⊥ )⊥ . For example, let C = hγ i i be a code of length 1 over R∞ for some i. Then C ⊥ = {0} and (C ⊥ )⊥ = R∞ since R∞ is domain. This means that C ( (C ⊥ )⊥ . We have the following proposition. Proposition 3.7. Let C be a linear code over R∞ . Then C = (C ⊥ )⊥ if and only if C has type 1k for some k. Proof. First we note that (C ⊥ )⊥ ⊆ C. If C is a linear code then by Lemma 3.4, the code C ⊥ is a linear code with type 1n−k for some k. This implies that (C ⊥ )⊥ has type 1n−(n−k) = 1k . Proposition 3.8. Assume the notations given above. Let C be a self-orthogonal code over R∞ . Then the code Ψi (C) is a self-orthogonal code over Ri for all i < ∞. Proof. We have that [v, w] = 0 for all v, w ∈ C since C is a self-orthogonal code over R∞ . This gives that n X l=1

vl w l ≡

n X

Ψi (vl )Ψi (wl ) ( mod γ i ) ≡ Ψi ([v, w]) ( mod γ i ) ≡ 0 ( mod γ i ).

l=1

Hence Ψi (C) is a self-orthogonal code over Ri . By Lemma 3.6, we know that for a γ-adic [n, k] code C of type 1k , C i = Ψi (C) is an [n, k] code of type 1k over Ri . In the following, we consider codes over chain rings that are projections of γ-adic codes. Note that C i  C i+1 for all i. Thus if a code C over R∞ of type 1k is given, then we obtain a series of lifts of codes as follows: C1  C2  · · ·  Ci  · · · Conversely, let C be an [n, k] code over F = Re /hγi = R1 , and let G = G1 be its generator matrix. It is clear that we can define a series of generator matrices Gi ∈ Mk×n (Ri ) such that Ψi+1 i (Gi+1 ) = Gi , where Mk×n (Ri ) denotes all the matrices with k rows and n columns over Ri . This defines a series of lifts Ci of C to Ri for all i. Then this series of lifts determines a unique code C such that C i = Ci . Let C be a γ-adic [n, k] code of type 1k , and G, H be a generator and parity-check matrices of C. Let Gi = Ψi (G) and Hi = Ψi (H). Then Gi and Hi are generator and parity check matrices of C i respectively. 10

Lemma 3.9. Assume the notations given above. Let i < j < ∞ be two positive integers, then (i) γ j−i Gi ≡ γ j−i Gj (mod γ j ); (ii) γ j−i Hi ≡ γ j−i Hj (mod γ j ). Proof. Let xl be the row vectors of Gi and yl be the row vectors of Gj . Since we have that Gi = Ψji (Gj ), this implies that xl ≡ yl ( mod γ i ). Thus γ j−i xl ≡ γ j−i yl ( mod γ j ). The proof of the (ii) is similar. Lemma 3.10. Let i < j < ∞ be two positive integers. Then (i) γ j−i C i ⊆ C j ; (ii) v = γ i v0 ∈ C j if and only if v0 ∈ C j−i ; (iii) Ker(Ψji ) = γ i C j−i . Proof. (i) Let v be an arbitrary codeword of C i , by Lemma 3.9(ii), we have that Hj (γ j−i v)T = γ j−i Hj vT ≡ γ j−i Hi vT ≡ 0 ( mod γ j ). This implies that γ j−i C i ⊆ C j . (ii) We know that γ i v0 ∈ C j ⇔ γ i Hj v0T ≡ 0 ( mod γ j ). By Lemma 3.9(ii), we have that γ i Hj = γ j−(j−i) Hj ≡ γ j−(j−i) Hj−i ≡ γ i Hj−i ( mod γ j ). This implies that γ i v0 ∈ C j ⇔ γ i Hj−i v0T ≡ 0 ( mod γ j ). Hence we have that γ i v0 ∈ C j ⇔ Hj−i v0T ≡ 0 ( mod γ j−i ) ⇔ v0 ∈ C j−i . (iii) By the definition of Kernel and (ii), we know that the vector v ∈ Ker(Ψji ) if and only if v ∈ C j and v = γ i v0 , where v0 ∈ C j−i . Thus the result follows. Remark 5. Lemma 3.10(iii) shows that the Hamming weight enumerator of the kernel Ker(Ψji ) is equal to the Hamming weight enumerator of C j−i . We now study weights of codewords in lifts of a code. Suppose i < j. By Lemma 3.10(i), we know that any weight of a codeword in C i is a weight of a codeword in C j . This implies that if v ∈ C i then there exists a w ∈ C j such that w H (w) = wH (v), where wH (·) denotes the Hamming weight of a vector. But in general the converse is not always true. We have the following theorem. Theorem 3.11. Let C be a γ-adic code. Then the following two results hold. (i) the minimum Hamming distance dH (C i ) of C i is equal to d = dH (C 1 ) for all i < ∞; (ii) the minimum Hamming distance d∞ = dH (C) of C is at least d = dH (C 1 ).

11

Proof. (i) Let v0 be a vector of C 1 with minimal Hamming weight d of C 1 . By Lemma 3.10(iii), we know that γ i−1 v0 is a codeword of C i with Hamming weight d. Hence dH (C i ) ≤ d for all i. Now we use induction on the index number i and assume that dH (C j ) = d for all j ≤ i. Suppose that dH (C i+1 ) < d and there is a non-zero vector v ∈ C i+1 such that i wH (v) < d. Then wH (Ψi+1 i (v)) ≤ wH (v) < d. Since we have that dH (C ) = d we must i+1 i have that Ψi+1 i (v) = 0 in C . This implies that v ∈ Ker(Ψi ). By Lemma 3.10(iii), we get that v = γ i v0 , where 0 6= v0 ∈ C 1 . This means that 0 < wH (v0 ) = wH (v) < d, which is a contradiction. (ii) If there exists a non-zero codeword v ∈ C such that wH (v) < d, then let N be a sufficiently large integer such that ΨN (v) 6= 0. We would have that wH (ΨN (v)) ≤ wH (v) < d. This is a contradiction. In the remainder of this section, we focus on MDS and MDR codes. The Singleton bound is generalized to codes over finite Frobenius rings. Since finite chain rings and principal ideal rings are all special class of finite Frobenius rings, we shall give some properties of codes over Ri for all i in the following. It is well known (see [10]) that for codes C of length n over any alphabet of size m dH (C) ≤ n − logm (|C|) + 1.

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Codes meeting this bound are called MDS (M aximal Distance Separable) codes. For a code C of length n over an finite Quasi-Frobenius ring R, Horimoto and Shiromoto (see [8]) define the following: rC = min{l | there exists a monomorphism C → Rl as R − modules}. If C is linear, then we have (see [8]) dH (C) ≤ n − rC + 1.

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Codes meeting this bound are called MDR (M aximal Distance with respect to Rank) codes. A linear code C over R is called free if C is isomorphic as a module to Rt for some t. This implies that if C is free then rC = rank(C). We have the following two theorems. Theorem 3.12. Let C be a linear code over R∞ . If C is an M DR or M DS code then C ⊥ is an MDS code. Proof. Assume C is a code of length n and rank k with dH (C) = n − k + 1. Then we know that C ⊥ is type 1n−k . Since R∞ is a domain, we get that any n − k columns of the generator matrix of C ⊥ are linearly independent. This gives that the minimum Hamming weight of C ⊥ is n − (n − k) + 1 = k + 1. Theorem 3.13. Let C be a linear code over Ri , and C˜ be a lift code of C over Rj , where j > i. If C is an M DS code over Ri then the code C˜ is an M DS code over Rj . 12

Proof. Assume C is a [n, k] code with minimum Hamming distance dH . We have that dH = n − k + 1 since C is an M DS code. Let v be a codeword of C such that wH (v) = dH . Then for any nonzero codeword v0 ∈ C, we have that wH (v0 ) ≥ wH (v). We know that C˜ is a [n, k] code, and that v can be viewed as a codeword of C˜ since we can write v = (v1 , · · · , vn ) where vl = al0 + al1 γ + · · · + ali−1 γ i−1 + 0γ i + · · · + 0γ j−1 . Let w be any lifted codeword of v. Then we have that wH (w) ≥ wH (v). On the other hand, for any lift codeword w0 of v0 , where v0 ∈ C, we also have that wH (w0 ) ≥ wH (v0 ) ≥ wH (v). This means that the minimum Hamming weight of C˜ is dH and this implies that C˜ is an M DS code for all j > i.

4

Self-Dual γ-adic Codes

In this section, we describe self-dual codes over R∞ . We fix the ring R∞ with R∞ → · · · → Ri → · · · → R2 → R1 and R1 = Fq where q = pr for some prime p and nonnegative integer r. The field Fq is said to be the underlying field of the rings. The following theorem can be found from [14]. Theorem 4.1. (i) If p = 2 or p ≡ 1 ( mod 4), then a self-dual code of length n exists over Fq if and only if n ≡ 0 ( mod 2); (ii) If p ≡ 3 ( mod 4), then a self-dual code of length n exists over Fq if and only if n ≡ 0 ( mod 4). Theorem 4.2. If i is even, then self-dual codes of length n exist over Ri for all n. i

Proof. Let C be the code with generator matrix G = γ 2 In . It is clear that C is selfi i i n n orthogonal over Ri since γ 2 γ 2 = γ i = 0 in Ri . We have that |C| = (q 2 )n = (q i ) 2 = |Ri | 2 . Therefore C is self-dual. Theorem 4.3. Let i be odd and C be a code over Ri with type 1k0 (γ)k1 (γ 2 )k2 · · · (γ i−1 )ki−1 . Then C is a self-dual code if and only if C is self-orthogonal and kj = ki−j for all j. Proof. We know that C ⊥ has type 1ki (γ)ki−1 (γ 2 )ki−2 · · · (γ i−1 )k1 . Hence the only if part follows. Now assume that C is a self-orthogonal code of length n and kj = ki−j for all j. Let l = b 2i c, where b c denotes the greatest integer function. Since i is odd, we have n=

i X j=0

i−1

kj = 2

2 X

j=0

13

kj = 2

l X j=0

kj .

(21)

n

Since C is self-orthogonal, C is self-dual if and only if |C| = (q i ) 2 . We have that i−1 i−1 i−1 i X X X X logq |C| = (i − j)kj = i kj − jkj = in − jkj = in − S, j=0

where S =

i P

j=0

j=0

j=0

jkj . By Equation (21), we have that

j=0

S =

i−1 X

jkj + i(n −

j=0

= in −

i−1 X

kj ) = in −

j=0

in 2

(i − j)kj

j=0

i i X X (i − j)ki−j = in − jkj = in − S. j=0

This implies that S =

i X

j=0

and logq |C| = in −

in 2

=

in . 2

Therefore C is self-dual.

Theorem 4.4. If C is a self-dual code of length n over R∞ then Ψi (C) is a self-dual code of length n over Ri for all i < ∞. Proof. Since C is a self-dual, we have that C = C ⊥ . This gives that C = C ⊥ = (C ⊥ )⊥ . By Proposition 3.7, the code C has type 1k for some k. Hence we have that k = n − k, this n gives that k = n2 . It is easy to get that rank(Ψi (C)) = n2 and so Ψi (C) has (pri ) 2 elements. By Proposition 3.8, Ψi (C) is self-orthogonal. Therefore Ψi (C) is a self-dual code. Corollary 4.5. Let C be a self-dual code of length n over R∞ . Recall that p is the characteristic of the underlying field F. We have (i) If p = 2 or p ≡ 1 ( mod 4), then n ≡ 0 ( mod 2); (ii) If p ≡ 3 ( mod 4), then n ≡ 0 ( mod 4). Proof. This result follows by Theorem 4.4 and Theorem 4.1. The following theorem gives a method to construct a self-dual code over F from a self-dual code over Ri . Theorem 4.6. Let i be odd. A self-dual code of length n over Ri induces a self-dual code of length n over Fq . Proof. Let C be a code over Ri of type generator matrix G as follows:  Ik0 A0,1 A0,2 A0,3   γIk1 γA1,2 γA1,3   γ 2 Ik2 γ 2 A2,3  G =  ...   ..  .  14

1k0 (γ)k1 (γ 2 )k2 · · · (γ i−1 )ki−1 with standard A0,i γA1,i γ 2 A2,i ... ..

.

..

. γ i−1 Iki−1 γ i−1 Ai−1,i

      .    

Let 

Ik0 A0,1 A0,2 A0,3   Ik1 A1,2 A1,3   Ik2 A2,3 ˜ =  G  .. .. . .   .. .. ..  . . .  Ikl

A0,i A1,i A2,i

      ,    

Al,i

˜˜ = Ψi (G) ˜ is a ( n ) × n matrix over Ri . Let G ˜ be the where l = b 2i c. By Equation (21), G 1 2 ˜˜ It is clear that matrix over Fq and let C˜˜ be the code over Fq with generator matrix G. ˜˜ = n , and thus it remains to show that C˜˜ is self-orthogonal. Let v00 , w00 be any two rank(C) 2

˜˜ suppose v00 = Ψi (v0 ) and w00 = Ψi (w0 ), where v = γ s v0 and w = γ t w0 are row vectors of G, 1 1 row vectors of G with s, t ≤ l. We have that 0 = [v, w] = [γ s v0 , γ t w0 ] = γ s+t [v0 , w0 ]. This implies that [v0 , w0 ] = 0 since s + t < i. In particular, the constant term in their inner product is zero. This means that [v00 , w00 ] = [v0 , w0 ] = 0. Theorem 4.7. Let R = Re be a finite chain ring, F = R/hγi, where |F| = q = pr , 2 6= p a prime. Then any self-dual code C over F can be lifted to a self-dual code over R∞ . Proof. Let G1 = (I | A1 ) be a generator matrix of C over R1 (= F). Since C is self-orthogonal, we have that I + A1 AT1 ≡ 0

(mod γ).

We show in the following by induction that there exist matrices Gi = (I | Ai ) such that T i T i Ψi+1 i (Gi+1 ) = Gi and I +Ai Ai ≡ 0 (mod γ ) for all i. Suppose we have that I +Ai Ai = γ Si . Let Ai+1 = Ai + γ i M , we want to find a matrix M such that I + Ai+1 ATi+1 ≡ 0

(mod γ i+1 ).

(22)

We know I + Ai+1 ATi+1 = I + Ai ATi + γ i (Ai M T + M ATi ) = γ i (Si + Ai M T + M ATi ). This gives that the matrix M should satisfy Si + Ai M T + M ATi ≡ 0 15

(mod γ).

(23)

In order to find all solutions to this equation, we consider the map η : Mn (F) → Mn (F) defined by η(M ) = Ai M T + M ATi . It is easily to get that η is linear and the kernel of η is Ker(η) = {KAi | where K is skew-symmetric} since Ai M T + M ATi = 0 if and only if (M ATi )T + M ATi = 0 if and only if M ATi = K is skew-symmetric if and only if M = K(ATi )−1 = −KAi . Note that Ai ATi = −I over F and gcd(2, p) = 1. This implies that 2 is a unit in F. Hence η(2−1 Si Ai ) = 2−1 (Ai Ati SiT + Si Ai ATi ) = 2−1 (−2)Si = −Si . Therefore the solutions to (22) exist and they are given by Ai+1 = Ai + γ i M, where M ≡ 2−1 (Si + K)A1 (mod γ) with any skew-symmetric K.

5

Codes over Principal Ideal Rings

In this section, we study codes over principal ideal rings. We study codes over this class of rings by the generalized Chinese Remainder Theorem. T Let R be a finite ring with a1 , · · · , as relatively prime ideals and si=1 ai = {0}. Let Φ : R → R/a1 × · · · × R/as

(24)

be the canonical R-module isomorphism. We denote the inverse isomorphism of Φ by CRT = Φ−1 : R/a1 × · · · × R/as → R. If Rj = R/aj , we denote R = CRT(R1 , · · · , Rs ). For j = 1, · · · , s, we let Cj be a code over Rj . Let C = CRT(C1 , · · · , Cs ) = Φ−1 (C1 , · · · , Cs ) = {Φ−1 (v1 , · · · , vs ) | vj ∈ Cj }. Then C is called the Chinese product of codes C1 , · · · , Cs . In the remainder of this section, we focus on codes over principal ideal rings. Let Re11 , Re22 , · · · , Ress be chain rings, where Rej j has unique maximal ideal hγj i and the nilpotency index of γj is ej . Let Fj = Rej j /hγj i. Let A = CRT(Re11 , · · · , Rej j , · · · , Ress ). We know that A is a principal ideal ring. For any 1 ≤ i < ∞, let Aji = CRT(Re11 , · · · , Rij , · · · , Ress ). 16

This gives that all the rings Aji are principal ideal rings. In particular, Ajej = A. We denote j , · · · , Ress ) by Aj∞ . CRT(Re11 , · · · , R∞ Let i1 , i2 be positive integers such that i1 < i2 . By the map Ψii21 in Equation (14), we get the following map which is also denoted by Ψii21 , Ψii21 : Re11 × · · · × Rij2 × · · · × Ress → Re11 × · · · × Rij1 × · · · × Ress , 1

j

s

1

(a , · · · , a , · · · , a ) 7→ (a , · · ·

, Ψii21 (aj ), · · ·

s

, a ).

(25) (26)

This gives the following commutative diagram Φ

Aji2 = CRT(Re11 , · · · , Rij2 , · · · , Ress ) −−−→ Re11 × · · · × Rij2 × · · · × Ress    Ψi2 i2 i2 −1 Φ ·Ψi ·Φ=CRT·Ψi ·Φy y i1 1 1 Φ

Aji1 = CRT(Re11 , · · · , Rij1 , · · · , Ress ) −−−→ Re11 × · · · × Rij1 × · · · × Ress . j , then Therefore the ring Aji1 can be viewed as a projection of Aji2 . If we replace Rij2 with R∞ we obtain the following chain

Aj∞ → · · ·

A A˜ k k j → Ajej → Aej −1 → · · · → Aj1 ,

where A˜ = CRT(Re11 , · · · , Fj , · · · , Ress ). For 1 ≤ i < ∞, let Cij be a code over Rij . Let Cij = CRT(Ce11 , · · · , Cij , · · · , Cess ) be the associated code over Aji . Let j j C∞ = CRT(Ce11 , · · · , C∞ , · · · , Cess )

be the associated code over Aj∞ . By the diagram above, we have the following diagram Φ

Cij2 = CRT(Ce11 , · · · , Cij2 , · · · , Cess ) −−−→ Ce11 × · · · × Cij2 × · · · × Cess    Ψi2 i i Φ−1 ·Ψi2 ·Φ=CRT·Ψi2 ·Φy y i1 1 1 Φ

Cij1 = CRT(Ce11 , · · · , Cij1 , · · · , Cess ) −−−→ Ce11 × · · · × Cij1 × · · · × Cess . This gives that the code Cij1 can be viewed as a projection of the code Cij2 . For all i < ∞, j we also have that the code Cij can be viewed as a projection of the code C∞ . We have the following theorems. j Theorem 5.1. Let C∞ be a linear code over Aj∞ . Then the following two results hold. (i) The minimum Hamming distance dH (Cij ) of Cij is equal to d = dH (C1j ) for all i < ∞; j j (ii) The minimum Hamming distance d∞ = dH (C∞ ) of C∞ is at least d = dH (C1j ).

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Proof. (i) By Lemma 2.8 in [3], we know that dH (Cij ) = dH (CRT(Ce11 , · · · , Cij , · · · , Cess )) = min{dH (Ce11 ), · · · , dH (Cij ), · · · , dH (Cess )}. Then for any i < ∞, by Theorem 3.11 (i), we have that dH (Cij ) = min{dH (Ce11 ), · · · , dH (Cij ), · · · , dH (Cess )} = min{dH (Ce11 ), · · · , dH (C1j ), · · · , dH (Cess )} = dH (C1j ). (ii) By Theorem 3.11 (ii), we have that j j dH (C∞ ) = min{dH (Ce11 ), · · · , dH (C∞ ), · · · , dH (Cess )} ≥ min{dH (Ce11 ), · · · , dH (C1j ), · · · , dH (Cess )} = dH (C1j ).

j be a self-dual code over Aj∞ . Then for any i < ∞, the code Cij is a Theorem 5.2. Let C∞ self-orthogonal code over Aji . j j , · · · , Cess is a self-dual code, we have that all the codes Ce11 , · · · , C∞ Proof. Since C∞ j are self-dual over Re11 , · · · , R∞ , · · · , Ress respectively. By Proposition 3.8, we know that Cij is self-orthogonal for all i < ∞. Therefore the code Cij is a self-orthogonal code over Aji for all i < ∞.

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[7] Hammons A. R. , Jr., Kumar P. V., Calderbank A. R., Sloane N. J. A., Sol´e P., The Z4 linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40, 1994, 301-319. [8] Horimoto, H. and Shiromoto, K., A Singleton bound for linear codes over quasiFrobenius rings, Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Hawaii (USA), 1999, 51–52. [9] Hungerford T.W., Algebra, Springer-Verlag, New York, 1974. [10] MacWilliams F. J., Sloane N. J. A., The Theory of Error-Correcting Codes. NorthHolland, Amsterdam, 1977. [11] McDonald B. R., Finite Rings with Identity, Marcel Dekker, Inc., New York, 1974. [12] Norton G. H., S˘al˘agean A., On the Hamming distance of linear codes over a finite chain ring, IEEE Trans. Inform. Theory, 46, 2000, 1060–1067. [13] Pless V. S., Huffman W. C., eds., Handbook of Coding Theory, Elsevier, Amsterdam, 1998. [14] Rains E., Sloane N.J.A., Self-dual codes, in the Handbook of Coding Theory, Pless V.S. and Huffman W.C., eds., Elsevier, Amsterdam, 1998, 177–294. [15] Wood J., Duality for modules over finite rings and applications to coding theory, Amer. J. Math., 121, 1999, 555–575.

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