A new class of superregular matrices and MDP convolutional codes

Abstract

This paper deals with the problem of constructing superregular matrices that lead to MDP convolutional codes. These matrices are a type of lower block triangular Toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular. We present a new class of matrices that are superregular over a sufficiently large finite field F. Such construction works for any given choice of characteristic of the field F and code parameters (n, k, δ) such that (n − k)|δ. Finally, we discuss the size of F needed so that the proposed matrices are superregular.

On Optimal Extended Row Distance Profile

“…First we introduce convolutional codes with finite support and in particular unit memory codes. We conclude this section by recalling the notion superregular matrices [2]. Such matrices have some similarities with the ones introduced in [3,7].…”

“…We have given conditions for the sliding generator matrix of a code to yield UM convolutional codes with nearly optimal extended row distances. A concrete construction have been presented based on a type of superregular matrices that had been recently used for the authors to build MDP [2]. Moreover, it was recently shown [15] that this class of matrices perform very well when considering rank metric instead of the Hamming metric, producing Maximum Sum Rank Distance convolutional codes.…”

“…If all the entries of G 0 and G (2) 1 are nonzero and the sliding generator matrix G r j is superregular then d r j ≥ (n − k) j + n + 1, i.e., C has an AOEDP.…”

A State Space Approach to Periodic Convolutional Codes

“…In recent years great effort has been dedicated to developing constructions of non-binary convolutional codes having good distance [2,9,14]. However, in contrast to block codes, the theoretical tools for the construction of convolutional codes with good designed distance have not been fully exploited.…”

Generalized column distances for convolutional codes

“…However, the problem of building MDP convolutional codes over small finite fields is still open and seems highly nontrivial (if possible at all). The two existing general constructions of MDP convolutional codes (see [15], [16], [17]) require far too large finite fields. Hence, one approach to this problem is to relax the MDP condition and try to find convolutional codes that are not MDP but posses good design column distance profile over small fields.…”