On Parity-Preserving Variable-Length Constrained Coding

Abstract

Previous work by the authors on parity-preserving fixed-length constrained encoders is extended to the variablelength case. Parity-preserving variable-length encoders are formally defined, and, to this end, Kraft conditions are developed for the parity-preserving variable-length setting. Then, a necessary and sufficient condition is presented for the existence of deterministic parity-preserving variable-length encoders for a given constraint. Examples are provided that show that there are coding ratios where parity-preserving variable-length encoders exist, while fixed-length encoders do not.

On Parity-Preserving Constrained Coding

“…To summarize, for the two different partitions, (3) and (4), of the alphabet Σ = {a, b, c, d}, Examples 7 and 8 present respective (capacity-achieving) parity-preserving variable-length encoders with a coding ratio of 1: the first encoder is deterministic, while the other is not. In fact, we show in [17] that for the partition (4), one cannot achieve a coding ratio of 1 by any deterministic parity-preserving variable-length encoder (unless one uses a degenerate base tag alphabet containing only even symbols).…”

“…In this section, we show through examples that the flexibility of having variable-length (tags and) labels strictly increases the attainable range of coding ratios, compared to the fixed-length case. A more thorough study of variable-length encoders is deferred to a subsequent work [17]. For more on variable-length encoders (in the nonparity-preserving setting), see [2], [4], [5], [6], [7], [8], [13, §6.4].…”

“…The purpose of this work is to initiate a study of parity-preserving encoders, starting with the special case of fixed-length encoders, where the window length over which the parity is preserved is a fixed multiple of p. We provide a formal definition of our setting in Section I-B below, preceded by some background and definitions which are taken from [13]. The more general variable-length model is a subject of future work [17] and is briefly discussed in Section IV.…”