Column Distances of Convolutional Codes Over <inline-formula> <tex-math notation="LaTeX">${\mathbb Z}_{p^r}$ </tex-math> </inline-formula>

Abstract

Maximum Distance Profile codes over finite non-binary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of p-encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) Maximum Distance Profile convolutional codes over Zpr is presented. I. INTRODUCTION Massey and Mittelholzer [19] showed that the most appropriate codes for phase modulation are the linear codes over the residue class ring Z M and this class includes the convolutional codes over Z M , where M is a positive integer. Fundamental results of the structural properties of convolutional codes over finite rings can be found, for instance, in [7] and [12]. Fagnani and Zampieri [7] studied the theory of convolutional codes over the ring Z p r in the case when the input sequence space is a free module. The problem of deriving minimal encoders (left prime and row-reduced) was posed by Solé et al. in [26] and solved by Kuijper et al. in [16] and [17] using the concept of minimal p-encoder, which is an extension of the concept of p-basis introduced in [29] to the polynomial context. The search for and design of good convolutional codes over Z p r have been investigated in several works in literature. Unit-memory convolutional codes over Z 4 that give rise to binary trellis codes with high free distances together with several concrete constructions of these codes were reported in [2] and [15]. In [13] two 16-state trellis codes of rate 2/4, again over Z 4 , were found by computer search. Also worth

Decoding of Z2S Linear Generalized Kerdock Codes

“…These results have led to an increased interest 16 in designing linear codes over finite rings. Several extensions of these codes to the Z 𝑝𝑆 17 (where 𝑝 is a prime and 𝑆 is a positive integer) were also developed [1,[8][9][10][11][12][13][14][15][16][17]. Solé [8] has 18 suggested in 1988 that 𝑝-adic cyclic codes should be investigated [9].…”

“…Several extensions of these codes to the Z 𝑝𝑆 17 (where 𝑝 is a prime and 𝑆 is a positive integer) were also developed [1,[8][9][10][11][12][13][14][15][16][17]. Solé [8] has 18 suggested in 1988 that 𝑝-adic cyclic codes should be investigated [9]. Generalization of 19 binary duadic codes to the setting of Abelian codes over the ring Z 2𝑆 was presented in [10].…”

“…Generalized Kerdock and Delsarte-Goethals codes over the ring Z 2 S (an extension of Z 4 linear Kerdock and Delsarte-Goethals codes to Z 2 S ) were introduced in [7]. Convolutional codes over rings were discussed in [1,8,9], and LDPC codes over rings were investigated in [10][11][12].…”