Constant-Weight Array Codes

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.