Abstract
In this paper, we study convolutional codes with a specific cyclic structure. By definition, these codes are left ideals in a certain skew polynomial ring. Using that the skew polynomial ring is isomorphic to a matrix ring we can describe the algebraic parameters of the codes in a more accessible way. We show that the existence of such codes with given algebraic parameters can be reduced to the solvability of a modified rook problem. It is our strong belief that the rook problem is always solvable, and we present solutions in particular cases.
Highlights
Convolutional codes (CC’s, for short) form an important class of error-correcting codes in engineering practice
It is easy to see that the existence of (n − 1)-dimensional σ-CCC’s with arbitrarily prescribed Forney indices implies the solvability of Problem 4.5 for any given numbers r1, . . . , rn−1 ∈ {0, . . . , n − 1}
We have studied σ-CCC’s in F[z]n where n | (q − 1) and where the automorphism σ induces a cycle of length n on the primitive idempotents of A
Summary
Convolutional codes (CC’s, for short) form an important class of error-correcting codes in engineering practice. In the 1970s, a lot of effort has been made to construct powerful CC’s with the aid of good block codes, see [14, 13] This idea has been resumed in the papers [24, 7]. In that case the skew polynomial ring A[z; σ] turns out to be isomorphic to a matrix ring over a commutative polynomial ring This allows us to construct generator polynomials of CCC’s with prescribed algebraic parameters. The existence of such codes reduces to a combinatorial problem followed by a problem of constructing polynomial matrices with certain degree properties. We close the paper with a short section illustrating how to generalize the results to codes that are cyclic with respect to a general automorphism
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