Constant-Weight Array Codes

TL;DR

This paper introduces constant-weight array codes (CWACs), a new class of codes that balance rate and decoding complexity, with constructions, decoding algorithms, and theoretical bounds linking them to existing code families.

Contribution

The paper proposes CWACs, generalizes classical bounds, and provides constructions and decoding algorithms, expanding the theoretical understanding of constant-weight and Hamming metric codes.

Findings

CWACs offer a tradeoff between rate and decoding complexity.

A concatenation-based construction and decoding algorithm for CWACs are provided.

Theoretical bounds for CWACs are generalized from classical code bounds.

Abstract

Binary constant-weight codes have been extensively studied, due to both their numerous applications and to their theoretical significance. In particular, constant-weight codes have been proposed for error correction in store and forward. In this paper, we introduce constant-weight array codes (CWACs), which offer a tradeoff between the rate gain of general constant-weight codes and the low decoding complexity of liftings. CWACs can either be used in the on-shot setting introduced earlier or in a multi-shot approach, where one codeword consists of several messages. The multi-shot approach generalizes the one-shot approach and hence allows for higher rate gains. We first give a construction of CWACs based on concatenation, which generalizes the traditional erasure codes, and also provide a decoding algorithm for these codes. Since CWACs can be viewed as a generalization of both binary constant-weight codes and nonrestricted Hamming metric codes, CWACs thus provide an additional degree of freedom to both problems of determining the maximum cardinality of constant-weight codes and nonrestricted Hamming metric codes. We then investigate their theoretical significance. We first generalize many classical bounds derived for Hamming metric codes or constant-weight codes in the CWAC framework. We finally relate the maximum cardinality of a CWAC to that of a constant-weight code, of a nonrestricted Hamming metric code, and of a spherical code.