Error Correcting Coding for a Non-symmetric Ternary Channel

TL;DR

This paper investigates error correcting codes for non-symmetric ternary channels, proposing a decoding method that simplifies ML decoding and characterizing code capabilities with bounds and optimality comparisons.

Contribution

It introduces a simplified decoding rule for non-symmetric ternary channels and analyzes code performance, including bounds and optimality, for the first time.

Findings

$ ext{d}_ ext{a}$-decoding is equivalent to ML decoding under certain conditions

Constructive bounds on code size are derived and compared to optimal codes

Proposed codes perform well and are sometimes optimal for short lengths

Abstract

Ternary channels can be used to model the behavior of some memory devices, where information is stored in three different levels. In this paper, error correcting coding for a ternary channel where some of the error transitions are not allowed, is considered. The resulting channel is non-symmetric, therefore classical linear codes are not optimal for this channel. We define the maximum-likelihood (ML) decoding rule for ternary codes over this channel and show that it is complex to compute, since it depends on the channel error probability. A simpler alternative decoding rule which depends only on code properties, called $\da$-decoding, is then proposed. It is shown that $\da$-decoding and ML decoding are equivalent, i.e., $\da$-decoding is optimal, under certain conditions. Assuming $\da$-decoding, we characterize the error correcting capabilities of ternary codes over the non-symmetric ternary channel. We also derive an upper bound and a constructive lower bound on the size of codes, given the code length and the minimum distance. The results arising from the constructive lower bound are then compared, for short sizes, to optimal codes (in terms of code size) found by a clique-based search. It is shown that the proposed construction method gives good codes, and that in some cases the codes are optimal.