Group convolutional codes

In this note we introduce the concept of group convolutional code. We make a complete classification of the minimal $S_3$-convolutional codes over the field of five elements by means of Jategaonkar's theorems.


Introduction
Block codes as left ideals in group algebras were introduced by S. D. Bermann in [1]. After that, several papers of MacWilliams, Landrock, Damgard, Lieber, Ward, Zimmermman and others gave more credit to this theory ( [3], [10], [11], [12], [13], [18]). In the context of convolutional codes, P. Piret [15], studied the H-codes, which can be seen as a generalized version of the group block codes in the convolutional case.
On the other hand, the concept of cyclic convolutional codes and their first properties were proposed by P. Piret and C. Roos in [14] and [16], respectively. More recently, H. Gluesing-Luerssen et al. ( [5], [6]) continue the study of cyclic convolutional codes. In the present paper, we give a definition of group convolutional code, which is a generalization of cyclic convolutional code and group block code. We introduce some important techniques in non-commutative algebra, concretely, the structure theorems of skew polynomials rings given in [9] by Jategaonkar.
The paper is organized as follows. In Section 2 we make the necessaries definitions related with convolutional codes we will use throughout the paper. Then we introduce the concept of group convolutional code and minimal one, this last will be the main object of our study since they are the building blocks for the rest of the codes. Next we summarize Jategaonkar's result on the structure of skew polynomial rings over semisimple rings, that we will use in the last section. Finally, Section 3 deals with the classification of the minimal S 3 -convolutional codes over the field of five elements. The isomorphism established between the skew polynomial ring and certain direct sums of rings of matrices over simplest skew polynomial rings will be crucial. Note that these codes are the smallest non-commutative group convolutional codes to consider. This result opens the way to consider more complicated examples.

Preliminaries and first results
Throughout this paper, IF denotes a finite field and n a positive integer such that the characteristic of IF , char(IF ), does not divide n. This assumption guarantees that for any group G of order n, the group algebra IF [G] is semisimple.
This paper deals with convolutional codes with additional algebraic structure. We adopt the following definition of convolutional code from [6].
Definition 1 A convolutional code of length n and dimension k is a direct summand C of IF [z] n of rank k as IF [z]-module.
Let r be a positive integer. Any matrix M ∈ M r×n (IF [z]) with rows given by a generating set of C as IF [z]-module is called generating matrix of the code C. If r = k, then M is called generator matrix or encoder of C.
The maximal degree of the k-minors of an encoder M is called the complexity of the code. A code of complexity zero is said to be a block code.
The free distance of a convolutional code is defined as follows. First, is the usual Hamming weight of the vector v i ∈ IF n . Then, the free distance of a convolutional code C ⊆ IF [z] n is defined as, We call (n, k, δ)-convolutional code a code with length n, dimension (or rank) k and complexity δ. We say that a (n, k, δ)-convolutional code with free distance d, C, is a MDS code (maximal distance separable) if d = S(n, k, δ), where S(n, k, δ) is the generalized Singleton bound, S(n, k, δ) = (n−k)(⌊ δ k ⌋+ 1) + δ + 1. For a given size field q, we have the so called Griesmer bound for convolutional codes over the field of q elements. It is defined as Here m denotes the maximum taken over the Forney indices of a (n, k, δ)convolutional code, and it is called the memory of the code. Also, IN denotes {1, 2, ...} if km = δ or {0, 1, 2, ...} if km > δ. A convolutional code over a field of q elements is said to be optimal if it reaches the Griesmer bound (see [7] ).
Let G = {g 1 , ..., g n } be a finite group of order n. We consider the group IF -algebra A = IF [G] and the IF -isomorphism β : IF n → A given by β(v 1 , ..., v n ) := n i=1 v i g i . On the other hand, we have the canonical isomorphism ψ : Now, let σ be an IF -automorphism of A and R = A[z; σ] be the skew polynomial ring. The multiplication rule in R is given by az = zσ(a) for all a ∈ A. The map ρ σ : IF [z] n → A[z; σ] defined just like ρ is the key for the next definitions ( in [5] essentially appears the respective definitions in the particular case of a cyclic group). Note that ρ σ is an isomorphism of left IF [z]-modules.
We will see that this definition coincides with the usual one where only is required that ρ σ (C) is a direct summand as IF [z]-module.
b) Let C be a (G, σ)-convolutional code. Then I = ρ σ (C) is a direct summand left ideal of R. If I is indecomposable then it is done. In the contrary case, I = I 1 ⊕ I 2 where I i is a nonzero left ideal of R for i = 1, 2. Again if both I i are indecomposable it is done. This procedure can be repeated and must stop since the ideal I has finite rank as IF [z]-module and the I i 's are free IF [z]-modules.
It is standard that any minimal (G, σ)-convolutional code is generated as left A[z; σ]-module by a primitive idempotent element of A[z; σ]. This paper mainly deals with the problem of finding these primitive idempotents. We are interested in the matrix approach of A[z; σ]. Next, we make an account of results on the interpretation of the elements of A[z; σ] as matrices in some matrix ring. We use Jategaonkar's results (cf. [9]) in order to give an explicit isomorphism of rings between A[z; σ] and the rings constructed via matrix rings.
For the rest of this section, let A be a finite ring (non necessarily commutative), σ : A → A be an automorphism and z an indeterminate. The skew polynomial ring R = A[z; σ] admits a variable change in z such that R is again a skew polynomial ring: let u be a unit in A and u the inner automorphism of A defined by u(a) = u −1 au, a ∈ A. It is easy to check that The following rings are intimately related to the skew polynomial rings. Let K be a ring and ρ : K → K an automorphism. Let D = K[x; ρ], m > 0 and P the subring of M m (D) consisting of all the matrices (d ij ) satisfying the next two conditions: (1) d ij ∈ D ∀i, j; (2) d ij ∈ xD if i > j. We denote the subring P by {K, m, ρ, x}. We also denote by I n the set {1, ..., n}.
We recall the concept of set of matrix units that appears is [8, P. 52]. Let A be a ring. A finite subset {e ij : i, j ∈ I n } in A is called set of matrix units in A if verifies the following two conditions: n i=1 e ii = 1 and e ij e kl = δ jk e il where δ jk is the Kronecker delta. In particular, e ij = 0 for all i, j ∈ I n . A The following fact will be used frequently in the next section. Let A be a semisimple finite ring and {f 1 , ..., f m } a complete set of semiprimitive idempotent elements in A. Assume that σ : A → A is an automorphism such that σ(f i ) = f π(i) where π is the cycle over I m given by π = (1 2...m). Let R = A[z; σ]. Then, by [9, Lemma 3.1], there exists a finite field K, an automorphism ρ : K → K and a positive integer n such that R ∼ = M n ({K, m, ρ, x}) for some indeterminate x.
Note that the above positive integer n is the cardinality of a complete set of matrix units in Af 1 .

S -convolutional codes
In this section we are going to determinate the minimal S 3 -convolutional codes over the field with five elements via Jategaonkar's theorems [9]. We fix the field with 5 elements IF 5 and let A = IF 5 [S 3 ]. The ring A is semisimple by Maschke Theorem. First, we calculate a complete set of primitive orthogonal idempotents elements of A by means of theory of Young diagrams (see [2, pg. 190]). The list of the four idempotent is the following: We consider two classes of IF 5 -automorphism of A attending to the feasible permutation that produces over the set {f 1 , f 2 , f 3 }. One class will be represented by the identity permutation and the other by the permutation (1 2). By [9,Theorem 3.3], two automorphisms that produce the same permutation also produce isomorphic skew polynomial rings. Moreover, we will prove later that they are isometric, in the sense that there is ring isomorphisms between them that preserve the weight of the elements. So we only take in our study the identity automorphism (for the identity permutation) and any automorphism σ ∈ Aut IF 5 (A) such that σ(f 1 ) = f 2 , σ(f 2 ) = f 1 and σ(f 3 ) = f 3 (note that any automorphism maps f 1 to f 1 or f 2 , f 2 to f 2 or f 1 and f 3 to f 3 ). (1 2) We begin with the second type of automorphism. We take the automorphism σ such that σ(I) = I, σ(1 2) = 4( 1 2) (1 3 2).

The case of the permutation
We study separately hg 1 and hg 2 .
Once we have completely described the isomorphisms φ 1 and φ 2 , we have the ring isomorphism ). This isomorphism will allow us to make calculations in S ⊕ M 2 (IF 5 [x]) and then to reflect them in A[z; σ]. We are interested in the S 3 -convolutional codes, these are obtained by means of the direct summands left ideals of A[z; σ]. Hence, we get the primitive idempotents of S and M 2 (IF 5 [x]), and then we apply φ −1 to them. Note that it is easy to see that any idempotent in S or M 2 (IF 5 [x]) is primitive.
We resume all the above by stating that any minimal S 3 -convolutional code corresponding to an idempotent of S has the basis a 2i z 2i and a 2i+1 z 2i+1 coprime (or, equivalently, f (z) and f (−z) coprime), or a 2i z 2i = 1 and a 2i+1 z 2i+1 = 0. Hence the complexity is always deg(f ). In both cases, these codes can be seen as codes of length 2 by concatenation.
For several small deg(f ) we can compute the free distance of some of these codes. For example, if f (z) = bz + a with a, b = 0 the code generated by (f (z), f (−z), f (−z), f (−z), f (z), f (z)) has free distance 12 and so is a MDS code. It is also easy to see that if f (z) = a + bz + cz 2 with a, b, c = 0, then the code generated by (f (z), f (−z), f (−z), f (−z), f (z), f (z)) has free distance 18 and so is a MDS code too. Now we focus our attention into the idempotents of M 2 (IF 5 [x]). Set d = φ −1 2 (B) = 4(1 2 3) + (1 3 2) ∈ Ag 2 . Note that d 2 = 2g 2 , hence d 2t = 2 t g 2 and d 2t+1 = 2 t dg 2 .
We consider an idempotent matrix in M 2 (IF 5 [x]): r s t 1 − r with r = 0, 1. Since r(1 − r) = ts, we call r = ah, s = hc, t = ab, 1 − r = bc. Then we have the following equalities: Then Hence the left ideals generated by r s t 1 − r and a c 0 0 are the same.
So we only have to transform a c 0 0 into an element of A[z; σ] and then calculate the associated convolutional code.
(Note that g 2 is the identity in Ag 2 ).

3.2
The case of the identity permutation Now we study the S 3 -convolutional codes that are obtained when the automorphism maps f 1 to f 1 . We can take, without lost of generality, σ = id A . Then Hence Ag 1 [z] has only two idempotents different from 0 and 1, concretely, ε 1 and ε 2 , which generate two direct summand left ideals of Ag 1 [z]. The S 3convolutional code associated to ε 1 has rank 1, a basis is { (1, 1, 1, 1, 1, 1)}, that is, it is a block code. The S 3 -convolutional code associated to ε 2 has also rank 1, a basis is {(1, 4, 4, 4, 1, 1)}, i.e., it is a block code too. These are the only minimal codes to consider in the component Ag 1 [z].
Next, we study the component Ag 2 [z]. In the same way that in the case σ = id A above, we find idempotent elements in Ag 2 [z] corresponding to the respective idempotent matrices in M 2 (IF 5 [z]).
We start with the same situation that in the case σ = id. We consider an arbitrary idempotent matrix r s t 1 − r with r(1 − r) = ts and r = 0, 1.
We will reach to the same conclusion that in the case σ = id: it is enough to work with the matrix a c 0 0 . Then, this matrix is performed into the element aε 3 + cε 34 of Ag 2 [z]. The associated generating matrices of the minimal S 3 -convolutional codes are obtained in a similar way to the case σ = id A : we only have to put in those matrices a 2 = c 2 = 0 and consider a 1 = a, c 1 = c as arbitrary coprime polynomials in IF 5 [z]. The generating matrix of the code has the following rows: Therefore the code has rank 2, {w 2 , w 3 } is a basis and the complexity is max{2deg(a), 2deg(c)}.
When r = 0 or r = 1, we can also reduce the matrices to reach out the above case and then we get some particular cases.

Conclusions
All the minimal S 3 -convolutional codes over IF 5 have the parameters (6, 1, t) or (6, 2, 2t) (t an arbitrary positive integer). If we compare this with the parameters of minimal Z Z 6 -convolutional codes (that is, σ-cyclic convolutional codes) we get the same result (see [5,Theorem 3.8]). Hence all minimal group convolutional codes of length 6 over the field of five elements have parameters (6, 1, t) or (6, 2, 2t). The positive integer t corresponds with the (constant) Forney indices of the code. Also note that general group codes are significantly more complicated than σ-cyclic convolutional ones. When σ = id, cyclic convolutional codes are always block codes, however, this is not the case for S 3 -convolutional codes. Finally, some free distances have been computed for these minimal S 3 -convolutional codes. The calculations show that MDS-convolutional codes (or optimal codes) appear frequently in this setting. It would be interesting to give some information on the free distance of group convolutional codes in terms of the algebraic structure of the groups.