Self-Dual Codes over Non-Commutative Frobenius Rings Steven T. Dougherty∗ Department of Mathematics University of Scranton Scranton, PA 18518 USA Andr´e Leroy Facult´e Jean Perrin Univerist´e d’Artois Lens, France June 25, 2014

Abstract We study self-dual codes over non-commutative Frobenius rings. It is shown that a code is equal to its left orthogonal if and only if it is equal to its right orthogonal. Constructions of self-dual codes are given over Frobenius rings that arise from selfdual codes over the center of the ring. These constructions are used to show for which lengths self-dual codes exist over various rings.

Keywords: Self-dual codes; non-commutative rings MSC Classification: 94B05 16D99

1

Introduction

Self-dual codes over rings and fields are an important and interesting class of codes. They have connections to unimodular lattices, invariant theory and design theory. See [5] for a description of these connections and [15] for a complete description of self-dual codes. ∗

The author is grateful to the Univerist´e d’Artois where he stayed while this work was completed.

1

Initially, self-dual codes were studied when the alphabet was a field. Self-dual codes over fields were used extensively in the study of designs, see [1] for numerous examples of this connection. Later self-dual codes over Z2k and Galois rings were studied especially in their relationship with unimodular lattices, see [2] for example. This leads to the study of self-dual codes over commutative Frobenius rings in [6]. In [6], it was shown when self-dual codes exist over commutative Frobenius rings by examining the decomposition of the ring via the Chinese Remainder Theorem. Numerous papers have been written studying self-dual codes over different commutative rings, see [10] and [17] for example. Important open questions concerning self-dual codes are given in [8]. Up to this point self-dual codes over non-commutative rings have received little attention. As one example, Hermitian self-dual codes over the non-commutative ring Σ2m = Z2k [i, j, k] were studied in [3]. However, in that ring, the fact that the inner-product was the Hermitian inner-product made the case much more similar to the commutative case in that there was a unique orthogonal to a code. In this work, we shall examine self-dual codes over noncommutative Frobenius rings.

1.1

Rings

All rings in the paper have a multiplicative identity and are assumed to be finite. Right and left R-modules are denoted by MR and R M respectively. As usual, the set of homomorphisms from the R-module M to the R module N is denoted by HomR (M, N ). Given a left (right) c = Hom (M, Q/Z) is a right (left) R-module. Note R-module M , the character module M Z c is defined as homomorphisms into C∗ instead, however that in the literature, sometimes M it is a simple matter to go from one of these definitions to the other. As first described in [16], we shall assume that all rings are Frobenius so that both MacWilliams theorems hold. For a finite ring, the following statements are equivalent: • R is Frobenius. b∼ • As a left module R =R R. b∼ • As a right module R = RR . If R is a finite-dimensional k-algebra, k a field, the above statements are also equivalent to the following ones: • There exists a nonsingular bilinear pairing B : R × R −→ k such that B(xy, z) = B(x, yz) for all x, y, z ∈ R. • There exists a hyperplane H in the k-space R which contains no nonzero right ideal.

2

See Example 4.4 in [16] for a list of examples of Frobenius rings. b having a generating character χ. This generating Frobenius rings are characterized by R character is vital in producing the MacWilliams relations for weight enumerators for codes over a Frobenius ring and is used extensively throughout the paper. Let R be a ring, the Jacobson radical J(R) consists of all annihilators of simple left R-modules. The Jacobson radical can be characterized as the intersection of all maximal right ideals. Notice that the Jacobson radical is a two sided ideal. For any ring R, we define the center of the ring Z(R) to be {α | α ∈ R, αβ = βα, ∀β ∈ R}. The center of a ring is a subring of the ambient ring but not an ideal. The socle of a ring R, Soc(R), is defined as the sum of all the minimal one sided ideals of the ring. For Frobenius rings, the sum of all the minimal left ideals is equal to the sum of all the minimal right ideals. So we can speak of a single socle of the ring. In this case, the socle is equal the left annihilator of the Jacobson radical.

1.2

Codes

Let R be a finite Frobenius ring. A code C of length n is a subset of Rn . If the code C is a left submodule of Rn then C is said to be left linear and if C is a right submodule then C is said to be right linear. Given a set of vectors v1 , v2 , . . . , vt ∈ Rn , their left span is X hv1 , v2 , . . . , vt iL = { αi vi | αi ∈ R} and their right span is hv1 , v2 , . . . , vt iR = {

X

vi αi | αi ∈ R}.

It is immediate that the left span is left linear and that the right span is right linear. Notice that hv1 , v2 , . . . , vt iL and hv1 , v2 , . . . , vt iR are not necessarily equal as in the following example. ! ! 1 0 a b Example 1. Let v = ∈ M2,2 (F2 ). Then hviR = { | a, b, ∈ F2 } and 1 0 a b ! a 0 hviL = { | a, b ∈ F2 }. We shall return to this example later to examine these b 0 codes more closely. P A set of vectors v1 , v2 , . . . , vt in Rn is left linearly independent if αi vi = 0 implies P αi = 0 for all i and it is right linearly independent if vi αi = 0 implies αi = 0 for all i. A code C is said to be a left (right) free code if it is a left (right) linear and a free module, that is, it has a basis, where a basis is a set of linearly independent vectors that generates 3

the code. In this case, the code is isomorphic to k direct copies of the ring R for some integer k as a left (right) module. For an arbitrary code there is no guarantee that there is a linearly independent generating set. In fact, in most cases there is not. Specifically, in any case where the code is not free there is not a linearly independent generating set. It is possible for a set of vectors to be left linear independent but not right linear independent and vice versa. Therefore we need to specify whether it is left or right free. However, in most applications in this paper the free codes will be both. Therefore, if the code is both left and right free then we shall simply say that it is free. For a code over an alphabet A = {a0 , a1 , . . . , as−1 }, the complete weight enumerator is the following polynomial in commuting indeterminates: cweC (xa0 , xa1 , . . . , xas−1 ) =

s−1 XY

xanii (c)

(1)

c∈C i=0

where there are ni (c) occurrences of ai in the vector c. The Hamming weight enumerator of a code C of length n is defined to be X WC (x, y) = xn−wt(c) y wt(c) , c∈C

where wt(c) = |{i | ci 6= 0}|. It is immediate that WC (x, y) = cwe(x, y, y, . . . , y).

2

Orthogonals

We define the following inner product on the ambient space Rn . For v = (vi ) and w = (wi ) define X [v, w] = vi w i . (2) The following lemma is immediate. Lemma 2.1. Let v, w ∈ Rn and α ∈ R then [αv, w] = α[v, w] and [v, wα] = [v, w]α. Notice that unlike for commutative rings if α multiplies v on the right or w on the left it does not necessarily come out of the inner product. For any code C in Rn define the following orthogonals: L(C) = {v ∈ Rn | [v, c] = 0, ∀c ∈ C}

(3)

R(C) = {v ∈ Rn | [c, v] = 0, ∀c ∈ C}.

(4)

and In the case when the ring is commutative the two orthogonals coincide and they are simply denoted by C ⊥ . The following lemma is well known and easily shown. 4

Lemma 2.2. Let C be a code, then L(C) is a left linear code and R(C) is a right linear code. Notice that L(C) is not necessarily right linear since vi αci + wi βci may not be equal to vi ci α + wi ci β. A characteristic of Frobenius rings is that they have a generating character. Namely, for b has a generating character χ, that is Rχ = R b = χR. We define the Frobenius rings R, R matrix T , where T is an |R| by |R| matrix given by: (T )a,b = (χ(ab))

(5)

where a and b are in R. The following is proven in [16]. Theorem 2.3. (Generalized MacWilliams Relations) Let R be a Frobenius ring, with |R| = k + 1. Let xi be the indeterminate that corresponds to the i-th element of R and let T be the |R| by |R| matrices defined by (T )a,b = (χ(ab)). If C is a left submodule of Rn , then cweC (x0 , x1 , . . . , xk ) =

1 cweR(C) (T t · (x0 , x1 , . . . , xk )). |R(C)|

If C is a right submodule of Rn , then cweC (x0 , x1 , . . . , xk ) =

1 cweL(C) (T · (x0 , x1 , . . . , xk )). |L(C)|

Similarly, for the Hamming weight enumerator we have the following. If C is a left submodule of Rn , then 1 WC (x, y) = WR(C) (x + (|R| − 1)y, x − y). |R(C)| If C is a right submodule of Rn , then WC (x, y) =

1 WL(C) (x + (|R| − 1)y, x − y). |L(C)|

The following lemma is well known, see [16]. Lemma 2.4. If C is a left linear code over a finite Frobenius ring, then |R(C)||C| = |R|n . If C is a right linear code then |L(C)||C| = |R|n . For finite fields this lemma says that the dimension of a code plus the dimension of its orthogonal is the dimension of the ambient space. For free codes the orthogonal of Rk is isomorphic to Rn−k . Therefore, the orthogonal of a free code is also a free code. Consider the code C = hviR given in Example ! 1. This code is a right linear code a a and has |C| = 4. We have that L(C) = { | a, b, ∈ F2 }. Hence |L(C)| = 4 and b b 5

! 0 0 |L(C)||C| = 16. Notice however that R(C) = { }. which has cardinality 1. Hence 0 0 the result of the lemma does not hold for both orthogonals, only those indicated in the conditions of the lemma. For codes over non-Frobenius rings the results of this lemma also do not hold. In that case, it is not possible to predict the size of the orthogonal simply by knowing the size of the code because the MacWilliams theorems do not hold. Lemma 2.5. If C is a left linear code over a Frobenius ring, then L(R(C)) = C. If C is a right linear code then R(L(C)) = C. Proof. Let C be a left linear code. If v ∈ R(C) then [C, v] = 0 and hence C ⊆ L(R(C)). Since C is left linear and R(C) is right linear, by Lemma 2.4, we have |R(C)||C| = |R|n = |L(R(C))||R(C)|. Therefore, |L(R(C))| = |C| and hence L(R(C)) = C. The second statement follows identically switching the roles of left and right. Lemma 2.5 can be seen as a generalization of the well-known double annihilator property for right (rep. left) ideals in a Frobenius ring (cf. Theorem 15.1 in [12]). The following corollary offers some properties of annihilators of general (not necessarily linear) codes contained in Rn . For subsets C and C 0 of Rn we denote by RC (resp. CR) the left (resp. right) submodule of Rn generated by C. Notice that L(C) = L(CR), R(C) = R(RC), and, if the zero vector belongs to both C and C 0 , we also have R(C + C 0 ) = R(C) ∩ R(C 0 ) and L(C + C 0 ) = L(C) ∩ L(C 0 ). Corollary 2.6. Let C and C 0 be subsets of Rn where R is a finite Frobenius ring. Then (a) L(R(C)) = RC and R(L(C)) = CR. (b) L(C) + L(C 0 ) = L(CR ∩ C 0 R) and R(C) + R(C 0 ) = R(RC ∩ RC 0 ). (c) R(C) ⊆ R(C 0 ) if and only if C 0 ⊆ RC. (c’) L(C) ⊆ L(C 0 ) if and only if C 0 ⊆ CR. Proof. In both (a) and (b) we prove only the first equality. The proof of the second equality is obtained in a similar way and is left to the reader. (a) The above remarks and Lemma 2.5 above show that L(R(C)) = L(R(RC)) = RC. (b) We have L(C) + L(C 0 ) = L(R(L(C) + L(C 0 ))) = L(R(L(C)) ∩ R(L(C 0 ))) = L(CR ∩ C 0 R). (c) and (c’) We only prove the statement (c). We have [C 0 , R(C)] ⊆ [C 0 , R(C 0 )] = 0. Hence, C 0 ⊆ L(R(C)) = RC. The converse statement is clear.

6

Let C be a left linear code. Then let D = R(C). By the previous lemma we have that L(D) = L(R(C)) = C. Hence any left linear code is of the form L(D) for some code D. Likewise any right linear code is of the form R(D) for some code D. Theorem 2.7. For a code C we have C ⊆ L(C) if and only if C ⊆ R(C). Proof. Consider the contrapositive. If C 6⊆ L(C) then there exists v ∈ C with v 6∈ L(C). This implies that there exists a vector w ∈ C with [v, w] 6= 0 since v 6∈ L(C). This gives that w 6∈ R(C), but w ∈ C. Hence C 6⊆ R(C). The proof for the second is identical except for using [w, v] instead of [v, w]. This allows us to define a code C as self-orthogonal if C ⊆ L(C), noting that this is equivalent to C ⊆ R(C). Theorem 2.8. Let v1 , v2 , . . . , vs be a set of s vectors. If [vi , vj ] = 0 for all i, j then hv1 , v2 , . . . , vs iL ⊆ L(hv1 , v2 , . . . , vs iR ) and hv1 , v2 , . . . , vs iR ⊆ R(hv1 , v2 , . . . , vs iL ). Proof. Assume [vi , vj ] = 0 for all i, j. Then using Lemma 2.1, we have [α1 v1 + α2 v2 + · · · + αs vs , v1 β1 + v2 β2 + · · · + vs βs ] = P P i,j αi [vi , vj ]βj = 0 i,j [αi vi , vj βj ] = Hence α1 v1 + α2 v2 + · · · + αs vs ∈ L(hv1 , v2 , . . . , vs iR ) and v1 β1 + v2 β2 + · · · + vs βs ∈ R(hv1 , v2 , . . . , vs iL ). Unlike for codes over commutative rings, simply because v and w are self-orthogonal vectors with [v, w] = 0 does not mean that hv, wiL is a self-orthogonal code. This is because [αv, βw] may not be 0 since the β does not come out of the ! inner-product. In 1 1 fact, this is often the case. For example in M2 (F2 ), if A = , then A2 = 0, but 1 1 ! ! ! 1 0 1 1 1 1 = is not self-orthogonal. In fact, it is idempotent. 0 0 1 1 0 0 We need something more to generate a self-orthogonal code, namely the following. Theorem 2.9. Let v1 , v2 , . . . , vs be a set of s vectors in Rn . Then [vi , αvj ] = 0 for all i, j and α ∈ R if and only if hv1 , v2 , . . . , vs iL is a self-orthogonal code. Proof. Assume [vi , αvj ] = 0 for all i, j then [α1 v1 +α2 v2 +· · ·+αs vs , β1 v1 +β2 v2 +· · ·+βs vs ] P P = i,j [αi vi , βj vj ] = i,j αi [vi , βj vj ] = 0. Hence hv1 , v2 , . . . , vs iL ⊆ R(hv1 , v2 , . . . , vs iL ). Then, by Theorem 2.7, the code is self-orthogonal. One technique which will be exploited later, is that the assumption of the previous theorem is true when the vectors vi are in Z(R)n .

7

3

Self-Dual Codes

In this section, we shall describe self-dual codes over Frobenius rings, and determine for which lengths self-dual codes exist for specific rings. Theorem 3.1. If C = L(C) then C = R(C). If C = R(C) then C = L(C). Proof. If C = L(C) then C is a left submodule and hence its left annihilator L(C) is a bimodule. That is, if v ∈ L(C) then [v, c] = 0 for all c ∈ C. For every r ∈ R, we have P P [vr, c] = (vi r)ci = vi (rci ) = [v, rc] = 0 (since C = L(C) is a left module). Hence vr ∈ L(C) so L(C) is a right R module. Then we have R(C) = R(L(C)) = C (since C is a right submodule). The proof of the other direction is identical switching the roles of the left and right orthogonals. An alternate proof of Theorem 3.1 can be given using Theorem 2.7. Namely, if C = L(C) then C ⊆ R(C). Then by a cardinality argument C = R(C). Definition 1. A code C is said to be self-dual if C = L(C) = R(C). It follows immediately from Theorem 3.1 that if C is a self-dual code over a Frobenius ring then C is both left linear and right linear. That is, it is a bimodule and hence the MacWilliams relations apply in both forms, meaning those given by the matrix T and the matrix T t . The following lemma is immediate. Lemma 3.2. A code C over a Frobenius ring R is self-dual if and only if it is self-orthogonal and |C|2 = |R|n . This leads to the following theorem. Theorem 3.3. If R is a Frobenius ring and |R| is not a square then self-dual codes exist only for even lengths. If C is a left (right) free self-dual code then its length must be even. Proof. Let n be the length of the code. By Lemma 3.2, for a self-dual code C we have |C||L(C)| = |C|2 = |R|n . Therefore, if |R| is not a square then n must be even. For a left (right) free code we have |C| = |R|k for some integer k. Then since the code is self-dual we have k = n2 and so n must be even. Simply because |R| is a square does not guarantee that there are self-dual codes of odd lengths. For example, consider the ring R = M2,2 (F2 ). Here |R| = 16, which is a square, however there are no self-dual codes of odd length over this ring. The next theorem is used to build self-dual codes from existing self-dual codes and will aid in determining for which lengths self-dual codes exist. 8

Theorem 3.4. If, over a Frobenius ring, C and D are self-dual codes of length n and m respectively , then C × D is a self-dual code of length n + m. If C and D are both left (right) free self-dual codes then C × D is a left (right) free self-dual code. Proof. The standard proof applies. Namely, if C and D are self-dual codes then if vi ∈ C, wj ∈ D, [(v1 , w1 ), (v2 , w2 )] = [v1 , v2 ] + [w1 , w2 ] = 0. Then |C × D| = |C||D| = n+m n m |R| 2 |R| 2 = |R| 2 . It is immediate that the product of free codes is a free code.

4

Self-Dual Codes of length 1

In this section, we shall study self-dual codes of length 1 over a Frobenius ring. These are important, first of all, because if there is a self-dual code of length 1, then by Theorem 3.4 there are self-dual codes of all lengths. Moreover, examining self-dual codes of length 1 helps to understand the structure of the ring. Since the rings are assumed to be finite, they are Artinian and hence their Jacobson radicals are nilpotent and thus contained in the nilradical. Hence these two radicals coincide in the finite case. Theorem 4.1. All self-dual codes of length 1 are two sided ideals contained in the Jacobson radical of the ring. Proof. For a self-dual code C we have that C 2 = 0 which implies that C is in the Jacobson radical. It follows immediately that if R is a semi-prime ring, that is N (R) = J(R) = {0}, then there are no self-dual codes of length 1. For example, there are no self-dual codes of length 1 if R is a field. Additionally, there are no self-dual codes for Mm,m (F ) of length 1, F a field, since J(Mm,m (F )) = Mm,m (J(F )) = {0}. Theorem 4.2. Let R be a finite Frobenius ring with Jacobson radical J(R). Let ` be such that J(R)` = 0, but J(R)h 6= 0 for h < `. If k ≥ d 2` e, then J(R)k is a self-orthogonal code. ` 1 ` If ` is even and |J(R) 2 | = |R| 2 then J(R) 2 is a self-dual code of length 1. Proof. If k ≥ d 2` e, then (J(R)k )2 = {0} and hence it is self-orthogonal. That is, it is ` 1 ` contained in its annihilator. If |J(R) 2 | = |R| 2 then J(R) 2 is self-orthogonal by the previous, ` ` ` ` and then since |J(R) 2 ||L((J(R) 2 ))| = |R| we have that J(R) 2 = (L(J(R) 2 )). If the Jacobson radical is a self-orthogonal code of length one then J(R)2 = 0 and hence J(R) is contained in its left (right) annihilator which is equal to the socle of R (the sum of all the minimal one sided ideals) (see 4.12 in [12]). We know that if C is a self-dual code 9

of length 1 then C ⊆ J(R), this implies that Soc(R) ⊆ C since Soc(R) is both the left and right annihiliator of J(R). Hence for a self-dual of length 1 we have Soc(R) ⊆ C ⊆ J(R).

(6)

This series of containments and the fact that L(J(R)) = Soc(R) gives the following result whose proof is elementary. Theorem 4.3. Let R be a finite Frobenius ring with Jacobson radical J(R). Then J(R) is a self-dual code if and only if J(R) = Soc(R) and in this case there are no other self-dual codes of length 1. ! a b Example 2. Let R be the finite ring R = { | a, b ∈ Fq } where σ is a fixed 0 σ(a) automorphism of the field Fq . This since the bilinear map defined by ! ring is Frobenius ! 0 0 a b a b β : R2 → R where β( , ) = ab0 + bσ(a0 ) is associative and non0 σ(a) 0 σ(a0 ) ! 0 b degenerate. The ring R has cardinality q 2 . The Jacobson radical J(R) = { |b∈ 0 0 Fq } has q elements and is self-orthogonal. Hence J(R) is a self-dual code of length 1 over this ring. Example 3. Consider the ring R as given on page 428 of [12], where R is the ring whose entries are of the form:   a x 0 0  0 b 0 0    (7)    0 0 b y  0 0 0 a where a, b, x, y ∈ Fq . This ring R has q 4 elements a detailed explanation as to why it is Frobenius. of the form  0 x 0  0 0 0    0 0 0 0 0 0

and is a finite Frobenius ring. See [12] for The Jacobson radical consists of matrices 0 0 y 0

    

(8)

and as such has q 2 elements. It is clear that J(R)2 = 0. Hence by the previous discussion J(R) is a self-dual code of length 1. A ring is called a left (right) chain ring if its lattice of left (right) ideals forms a chain. See [9] and the references therein for a description of self-dual codes over chain rings. If R is a finite chain ring then R has a unique minimal ideal. Then by Corollary 2.2 in 10

[11], we have that R is Frobenius and local. Hence all finite chain rings (commutative or non-commutative) are necessarily finite. For finite chain rings, if there is a self-dual code of length 1 it must be a power of the Jacobson radical, since these are the only ideals. For example, consider R = F4 [x, θ]/hx2 i where θ maps an element in the ground field to its square. This ring is a non-commutative chain ring and has J(R) = hxi which is self-dual code of length 1. We explain precisely when a power of the Jacobson radical is a self-dual code for chain rings in Theorem 4.4. As a commutative example consider Z16 , here J(R) = h2i and J(R)2 is a self-dual code of length 1. It is not true however, that all self-dual codes are powers of the Jacobson radical. Consider the principal ideal ring Z144 . We have J(Z144 ) = h6i and C = h12i is a self-dual code of length 1 which is not a power of the Jacobson radical. In fact, a self-dual code does not have to be in the chain of powers of the Jacobson radical. Consider, for example, the ring R = F2 [u1 , u2 , u3 ]/hu2i = 0, ui uj = uj ui i. This commutative ring has J(R) = hu1 , u2 , u3 i and J(R)2 = hu1 u2 , u1 u3 , u2 u3 i. The code C = hu1 + u2 + u3 i is a self-dual code of length 1 with C ⊂ J(R) but J(R)2 6⊂ C. It is also possible for a ring to have more than one self-dual code of length 1. Consider the ring Z2 [i, j, k] where the i, j, k satisfy the quaternion equations with the usual inner-product. Then h1 + ii, h1 + ji, and h1 + ki are distinct self-dual codes of length 1. Notice that this ring is commutative so we need not indicate whether we are generating on the left or the right. It is known, see [4], that a left finite chain ring is a right chain ring. Moreover, if the Jacobson radical of the finite chain ring R is J(R) then every left (right) ideal is of the form J(R)i for some i. Since the ring is finite we have that J(R)e = {0} for some minimal e. This index e is called the index of nilpotency. Theorem 4.4. Let R be a finite chain ring with index of niloptency e. If e is even then e J(R) 2 is a self-dual code of length 1. e

Proof. We have that (J(R) 2 )2 = {0}. Hence the code is self-orthogonal. But its dual must be of the form J(R)i for some i. This index i cannot be greater than 2e since the ideal is e already annihilated by J(R) 2 and the index i cannot be less than 2e since J(R)k 6= {0} for k < e.

5

Constructions of Self-Dual Codes

In this section, we shall give constructions of self-dual codes. Theorem 5.1. Let R be a finite Frobenius ring such that there exist x and y in Z(R) with x2 + y 2 = 0 and annR (x, y) = {0}. Then there exists free self-dual codes of all even lengths over R. 11

Proof. The code h(x, y)i is self-orthogonal since x2 + y 2 = 0 and the size of the code is |R| since annR (x, y) = {0}. Therefore h(x, y)i is a self-dual code of length 2. Notice that the code is both left and right linear. Then take direct products of C to get self-dual codes of all even lengths. Finally, we notice that the form of the generator matrix of length 2 indicates that this code is free. The code formed by taking direct products is free because the direct product of free codes is a free code. Any finite Frobenius ring that has an element α ∈ Z(R) such that α2 = −1 satisfies the conditions of Theorem 5.1 with x = 1 and y = α. Example 4. Let us consider the ring Z5 [i, j, k], where i, j, k satisfy the quaternion equations. This central simple algebra is in fact a Frobenius algebra. A nonzero degenerate associative bilinear form is given by B(x, y) = x¯ y +¯ xy, where x0 + x1 i + x2 j + x3 k = x0 −x1 i−x2 j−x3 k. We have that (1, 2) generates a self-dual code of length 2, since 22 = −1 and 2 is in the center of the ring. Notice that in the previous proof it was imperative that the element α of R was in the center of the ring. That is, simply because if there is an element α with α2 = −1, the vector (1, α) may not generate a self-dual code if α is not in the center since [(1, α), β(1, α)] may not be 0. For example, consider the ring F7 [i, j, k] where i, j, k satisfy the quaternion equations. Then [(1, i), j(1, i)] = j + i(−k) = j + j 6= 0. Hence (1, i) does not generate a self-dual code of length 2. Corollary 5.2. Let F be a finite field of characteristic 2 or characteristic 1 (mod 4). Then there exist self-dual codes of all even lengths in F G where G is any finite group. Proof. Let us first remark that it is well-known that F G is a Frobenius algebra. If F is a finite field of characteristic 2 or characteristic 1 (mod 4) then there exists a γ ∈ F such that γ 2 = −1. The vector (1, γ) ∈ F (G)2 generates a self-dual code of length 2 as in the previous theorem. Corollary 5.3. Let F be a finite field of characteristic 2 or characteristic 1 (mod 4). Then there exist self-dual codes of all even lengths over Mm,m (F ) for all m. Proof. As in the previous corollary we have γ ∈ F such that γ 2 = −1. Let Im be the m × m identity matrix. Then the element γIm satisfies the conditions in the previous theorem, that 2 is (γIm )2 + Im = 0. Of course, other subrings of matrix rings can have a similar construction without the base field having a square root of −1 as in the following corollary.

12

Corollary! 5.4. Let F be any field and let R be the ring of matrices in M2,2 (F ) of the form a b . Then there exist self-dual codes for all even lengths. −b a Proof. Notice that the ring R is a commutative Frobenius ring. (The dimension of R as an F vector space is 2, hence F itself ! is a hyperplane that does not contain any nonzero ideal). 0 −1 Consider the element in R. This element is a central square root of −1 and 1 0 hence satisfies the conditions of Theorem 5.1. The following theorem is similar to Theorem 5.1 except that it constructs self-dual codes of lengths a multiple of 4. Theorem 5.5. Let R be a finite Frobenius ring such that there exists x, y and z in Z(R) with x2 + y 2 + z 2 = 0 and annR (x, y, z) = {0}. Then there exist free self-dual codes of all lengths congruent to 0 mod 4 over R. Proof. Consider the following generator matrix: x 0 y z 0 x −z y

! .

The generated code is self-orthogonal since x2 + y 2 + z 2 = 0 and because the two rows are orthogonal with elements in the center of the ring. The size of the code is |R|2 since annR (x, y, z) = {0}. Therefore the code is a self-dual code of length 4. Then by taking direct products we have self-dual codes of all lengths congruent to 0 (mod 4). Notice that the code is both left and right linear. Then as in the case of length 2 the codes are free. Notice that in the previous theorem if there exists α, β ∈ Z(R) such that α2 + β 2 = −1. Then 1, α, β satisfy the conditions of the theorem and the form of the generator matrix is: ! 1 0 α β . 0 1 −β α Corollary 5.6. Let F be a finite field of characteristic 3 (mod 4). Then there exist self-dual codes of all lengths congruent to 0 (mod 4) in F G where G is any finite group. Proof. If F is a finite field of characteristic 3 (mod 4) then there exists γ, δ ∈ F such that γ 2 + δ 2 = −1. The vectors (1, 0, γ, δ) and (0, 1, δ, γ) ∈ F (G)2 generate a self-dual code of length 4 as in the previous theorem. Corollary 5.7. Let F be a field of characteristic 3 (mod 4). Then there exist free self-dual codes of all lengths congruent to 0 mod 4 over Mm,m (F ) for all m. 13

Proof. If F is a field F of characteristic 3 (mod 4) then there exists γ, δ ∈ F such that γ 2 + δ 2 = −1. Let Im be the m × m identity matrix. Then γIm and δIm take the role of y and z and Im = x in the previous theorem.

6

Constructions of Self-Dual Codes with the Center of the Ring

In this section, we give constructions of self-dual codes which use self-dual codes over commutative rings and the center of the ring. Note that the center of a ring is a subring of that ring and the ring is an algebra over its center. Lemma 6.1. Let C be a self-orthogonal code over a subring A of Z(R), that is C ⊆ An , where R is a ring. Let C 0 = hCiL over R. The C 0 is a self-orthogonal code in Rn . P P Proof. Let w = si=1 αi vi , vi ∈ C and w0 = tj=1 βj uj , uj ∈ C. Then we easily check that P P P P P P [w, w0 ] = [ si=1 αi vi , tj=1 βj uj ] = tj=1 si=1 [αi vi , βj uj ] = tj=1 si=1 αi [vi , uj ]βj = 0 Thus the code is self-orthogonal. Theorem 6.2. Let C be a self-dual code of length n over a subring A of Z(R), that is C ⊆ An , where A and R are Frobenius rings. If C has a generator matrix of the form G = (I n2 |B), then hGiL is a free self-dual code. Proof. By Lemma 6.1, the code D = hCiL will be self-orthogonal over R, i.e. D ⊆ L(D), where L(D) is the left orthogonal in Rn . Since G is of the form G = (I n2 |B), namely with the identity matrix of size n2 as the left n side of the matrix, we have that G generates a free code with cardinality |R| 2 . Because the elements of the generating matrix are in the center of the ring the code is both left and right linear. Theorem 6.3. Let R and A ⊆ Z(R) be finite Frobenius rings where R is a free module over A. If C is a self-dual code over A then hCiL is a self-dual code. Proof. By Lemma 6.1, the code D = hCiL will be self-orthogonal over R, i.e. D ⊆ L(C). We know that |C||L(C)| = |C|2 = |A|n . The ring R is a free algebra over A, hence we have that |R| = |A|k for some k. This implies that |hCiL | = |C|k . This gives that |hCiL |2 = (|C|k )2 = (|C|2 )k = |An |k = (|A|k )n = |R|n . Hence the code hCiL is self-dual. Example 5. Consider the matrix ring Mn,n (Zm2 ). The center of this ring consists of the diagonal matrices. The code C generated by mIn is a self-dual code over the center of length 1 and cardinality m. Then hCi is a self-dual code over Mn,n (Zm2 ). 14

Notice that not all self-dual codes are constructed from codes over the center. For example, the self-dual code in Example 2 does not have a generator of this form. Moreover, the ring has characteristic p where q = pe , p a prime, and Zp does not have self-dual codes of length 1 as in this example. Corollary 6.4. Let R be a finite Frobenius ring of characteristic k. If there exists a free self-dual code of length n over Zk then there exists a free self-dual code of length n over R. Proof. If the ring R has characteristic k, then the ring R contains Zk as a subring. Moreover, Zk is a subring of Z(R) and is Frobenius. Then by Theorem 6.3 we have the result. The existence of free self-dual codes over Zk is given in [7], we use these to obtain the following corollary to Theorem 6.3. Q Corollary 6.5. Let R be a finite Frobenius ring of characteristic k. Let k = pei i where pi is a prime and pi 6= pj when i 6= j. If pi ≡ 1 (mod 4) for all i then there are free self-dual codes of all even lengths and if there exists pi with pi ≡ 3 (mod 4) then there exist free self-dual codes of all lengths 0 (mod 4). For commutative rings, self-dual codes are constructed via the Chinese Remainder Theorem. Namely, any Frobenius ring is the product of local rings and any principal ideal ring is the product of chain rings. Then self-dual codes are constructed as products of self-dual codes over the component rings. While non-commutative rings do not necessarily decompose like this we can consider the direct sum of rings. It is known that the direct sum of Frobenius rings is Frobenius. Theorem 6.6. Let Ci be a self-dual code over the Frobenius ring Ri , for i = 1, . . . , k, of length n. Let R = ⊕Ri . Then C = ⊕Ci is a self-dual code over R of length n. Proof. The fact that the code C is self-orthogonal is immediate since the isomorphism beQ tween between R and ⊕Ri sends 0 to 0 in each ring. Then we have that |R| = |Ri |, Q 2 n 2 n |C| = |Ci | and |Ci | = |Ri | . This gives that |C| = |R| and then C is self-dual.

7

Building Up Constructions

One of the most important tools in the study of self-dual codes is the building up construction. See [6] for its application to commutative Frobenius rings. We shall generalize these constructions to the non-commutative case. We begin by generalizing the standard building up theorem. Theorem 7.1. Let R be a finite Frobenius ring such that there exists c ∈ Z(R) with c2 = −1. Let C = hv1 , v2 , . . . , vk iL be a self-dual code of length n over R. Let x be a vector in Z(R)n 15

with the property that [x, x] = −1. Let αi = [x, vi ]. Construct new vectors of length n + 2 by wi = (−αi , cαi , vi ) and w0 = (1, 0, x). Then C = hw0 , w1 , w2 , . . . , wk iL is a self-dual code of length n + 2. Proof. For i, j > 0, we have [βwi , γwj ] = βαi γαj + c2 βαi γαj + [vi , vj ] = βαi γαj − βαi γαj + [vi vj ] = 0. For w0 to generate self-orthogonal vectors, we need [βw0 , γw0 ] = 0. But [βw0 , γw0 ] = βγ + [βx, γx]. If x ∈ Z(R)n then this is 0. The remaining inner-product to consider is [βw0 , γwi ] where i > 0. Here we have [βw0 , γwi ] = −βγαi + [βx, γvi ] X = −βγαi + β( xj γ(vi )j ) X = −βγαi + β( γxj (vi )j ) = −βγαi + βγαi = 0. Hence the code C is self-orthogonal. P Let us now show that |C| = |R||C|. Notice that C = Rw0 + ki=1 Rwi . Since obviously P |Rw0 | = |R|, it is enough to prove that | ki=1 Rwi | = |C|. Let p denote the projection on the P P last n components from Rn+2 onto Rn . Then p(wi ) = vi and p( ki=1 Rwi ) = ki=1 Rvi = C. P P P Suppose that ki=1 γi wi ∈ ker(p). We then have that 0 = p( ki=1 γi wi ) = ki=1 γi vi . First notice that since x ∈ Z(R)n we get k X

γi αi =

i=1

k X

γi [x, vi ] = [x,

i=1

k X

γi vi ] = 0.

i=1

We then get k X i=1

γi wi = (−

k X i=1

k k X X γi αi , c( γi α)i , γi vi ) = (0). i=1

i=1

Pk

P This shows that p induces a bijection between i=1 Rwi and C = ki=1 Rvi . Since C is self¯ 2 = (|R||C|)2 = |R|2 |R|n = |R|n+2 . dual and R is Frobenius we have |C|2 = |R|n and so |C| This yields the result. We can generalize the second building up theorem as well. Theorem 7.2. Let R be a finite Frobenius ring such that there exists c, d ∈ Z(R) with c2 + d2 = −1. Let C = hv1 , v2 , . . . , vk iL be a self-dual code of length n over R. Let x, y be vectors in Z(R)n with the property that [x, x] = [y, y] = −1, [x, y] = 0. Let αi = [x, vi ] and βi = [y, vi ]. Let wi = (αi , βi , −cαi − dβi , −dαi + cβi , vi ), w−1 = (1, 0, 0, 0, x1 ) and w0 = (1, 0, 0, 0, x2 ). Then C = hw−1 , w0 , w1 , w2 , . . . , wk iL is a self-dual code of length n + 4. 16

Proof. The fact that C is self-orthogonal follows exactly as in Theorem 7.1. Then we see, as in the previous proof, that |C| = |R|2 |C| and then |C|2 = |R|n+4 and the code is self-dual. Note that it is imperative that the vectors x, x1 and x2 have elements in the center. It is because of this that one cannot say immediately that all codes can be constructed in this manner since a code may not have an element in its generator set that is contained in Z(R)n . As an example, consider any principal ideal that is self-dual that is not contained in the center of the ring.

8

Characters and Invariants

b as bimodules. Note that A ring is said to be a symmetric ring if R is isomorphic to R all commutative rings are symmetric. Let us mention the following classical example of a Frobenius ring that is not symmetric. Consider any field K and divide the free algebra Khx, yi by the ideal generated by X 2 , Y 2 and XY − λY X . It is easy to verify that this algebra is Frobenius and it is symmetric if and only if λ = 1. Notice also that this algebra is finite if K is finite. Lemma 8.1. Let χ be the generating character for a Frobenius ring R. The ring R is bR is symmetric ring if and only if the generating character χ = Ψ(1), where Ψ : R RR → R R the R-bimodule isomorphism has the property that χ(ab) = χ(ba) for all a, b ∈ R. Proof. For all a, b, we have that a · χ · b = Ψ(a(1)b) = Ψ(ab) = χ · ab. Apply this to 1 we have (a · χ · b)(1) = χ(ba) and χ · ab(1) = χ(ab). The converse implication is proved similarly. Example 6. Let p be a prime number. The ring R = Mn,n (Fp ) is symmetric with generating character χ defined by χ(A) = µ(T r(A)) , where µ : Fp → {0, 1, 2, . . . , p − 1} is the natural p bijection. Then since T r(AB) = T r(BA) we have that χ(AB) = χ(BA). Example 7. Let Fp be the finite field of order p and let G be a finite group. Consider the P group ring Fp G. Then every element is of the form ag g where ag ∈ Fp . The character is P defined as χ( ag g) = a1G . Then X X X X X X χ(( ag g)( bg g)) = ag bg−1 = bg ag−1 = χ(( bg g)( ag g)). Let χ be the generating character for a Frobenius ring R. Let H be an additive subgroup of Rn . Define the following: A(H) = {v ∈ Rn | χ(v · h) = 0, ∀h ∈ H} B(H) = {v ∈ Rn | χ(h · v) = 0, ∀h ∈ H}. P Note that χ(v · h) = χ(vi hi ). 17

Lemma 8.2. If R is a symmetric ring, then for any additive subgroup H of R we have that A(H) = B(H). Proof. By Lemma 8.1 we have that χ(xi hi ) = χ(hi xi ) and so the result follows. Theorem 8.3. Let C be a code over a symmetric Frobenius ring R, that is both left and right linear, i.e. a bimodule. Then L(C) = R(C). Proof. By Theorem 7.7 in [16], we have that L(C) = A(C) and R(C) = B(C). Then by Lemma 8.2 we have that L(C) = R(C). Hence, when the ring is symmetric, bimodules have a unique orthogonal. Theorem 8.4. Let C be a left linear code and D be a right linear code over a symmetric ring R. If cweC (xa0 , xa1 , . . . , xas−1 ) = cweD (xa0 , xa1 , . . . , xas−1 ) then cweR(C) (xa0 , xa1 , . . . , xas−1 ) = cweL(D) (xa0 , xa1 , . . . , xas−1 ). If WC (x, y) = WD (x, y) then WR(C) (x, y) = WL(D) (x, y). Proof. Since the ring R is symmetric, by Lemma 8.1 we have that T = T t . Then by Theorem 2.3 the weight enumerators of the respective orthogonals are the same. Note that if the rings in the previous theorem were not symmetric then the conclusion of the theorem may not be true. That is simply because two codes have the same weight enumerator does not mean that their left and right duals have identical weight enumerators. We shall describe how the theory of invariants can be applied in this case. If C is a self-dual code then C is a bimodule and is equal to both L(C) and R(C). This gives the following result. Theorem 8.5. The complete weight enumerator of a self-dual code over R is held invariant by both √1 T and √1 T t . |R|

|R|

Proof. A self-dual code is both left and right linear, hence applying the MacWilliams relations to the complete weight enumerator as both a left and right linear code produces the same weight enumerator. Hence the weight enumerator is held invariant by both √1 T and |R|

√1 T t . |R|

In the symmetric case the matrices T and T t are identical, but in the non-symmetric case they can be different. Notice that if C is a left linear code then L(R(C)) = C. This implies that √1 T √1 T t |R|

|R|

is a permutation matrix (it need not be simply the identity). The case for the Hamming weight enumerator is the same as it is for any commutative ring of order |R| since the MacWilliams relations are the same. Namely, the Hamming weight enumerator is held invariant by the matrix ! 1 1 |R| − 1 p . −1 |R| 1 18

This case is completely explained in Chapter 19 of [13].

References [1] E.F. Assmus, J.D. Key, Designs and their codes, Cambridge Tracts in Mathematics, 103. Cambridge University Press, Cambridge, 1992. [2] E. Bannai, S.T. Dougherty, M. Harada, M. Oura, Type II codes, even unimodular lattices, and invariant rings. IEEE Trans. Inform. Theory, Vol. 45, No. 4, 1999, 1194 1205. [3] Y.J. Choie, S.T. Dougherty, Codes over Σ2m and Jacobi Forms over the Quaternions, Appl. Algebra. Engr. Com. Comput. 15, No. 2, 2004, 129 - 147. [4] W. E. Clark, D. A. Drake, Finite chain rings. Abhandlungen aus dem mathematischen Seminar der Universit¨at Hamburg, 39, 1974, 147-153. [5] J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, NY 3rd ed., 1998. [6] S.T. Dougherty, J.L. Kim, H. Kulosman, Hongwei Liu, Self-Dual codes over commutative Frobenius Rings, Finite Fields and their Applications, Vol. 16, January 2010, 14 26. [7] S.T. Dougherty, M. Harada, P. Sol´e, Self-Dual Codes over Rings and the Chinese Remainder Theorem, Hokkaido Math Journal, Vol. 28, No. 2, 1999, 253 - 283. [8] S.T. Dougherty, J.L. Kim, P. Sol´e, Open Problems in Coding Theory, to appear in Contemporary Mathematics. [9] S.T. Dougherty, J.L. Kim, H. Liu, Constructions of self-dual codes over chain rings, International Journal of Information and Coding Theory, vol. 1, no. 2, 171 - 190. [10] S.T. Dougherty, B. Yildiz, Bahattin, S. Karadeniz, Self-dual codes over Rk and binary self-dual codes. Eur. J. Pure Appl. Math. 6 (2013), no. 1, 89 - 106. [11] X. Hou, A. Nechaev, A construction of finite Frobenius rings and its application to partial difference sets, J. Algebra, 309, 2007, 1 - 9. [12] T.Y. Lam, Lectures on Modules and Rings, Springer-Verlag, 1999. [13] F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes, NorthHolland, Amsterdam, 1977. 19

[14] V.S. Pless, W.C. Huffman, eds., Handbook of Coding Theory, Elsevier, Amsterdam, 1998. [15] E. Rains, N.J.A. Sloane, Self-dual codes, in the Handbook of Coding Theory, Pless V.S. and Huffman W.C., eds., Elsevier, Amsterdam, 1998, 177 - 294. [16] J. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math., Vol. 121, 1999, 555 - 575. [17] B. Yildiz, S. Karadeniz, Self-dual codes over F2 + uF2 + vF2 + uvF2 , J. Franklin Inst. 347 (2010), no. 10, 1888 - 1894.

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