On Parity-Preserving Constrained Coding

Abstract

Necessary and sufficient conditions are presented for the existence of fixed-length parity-preserving encoders for a given constraint. It is also shown that under somewhat stronger conditions, the stethering method guarantees an encoder that has finite anticipation.

On Parity-Preserving Variable-Length Constrained Coding

“…One possible parity-preserving tag assignment to the edges of E is shown in Table II. Similarly to the partition (1), it was shown in [14] that for the partition (3), too, one cannot achieve a coding ratio of 1 by any paritypreserving fixed-length encoder for S. The encoder in Figure 6 can be obtained from (an untagged copy of) the encoder in Figure 2 by replacing the outgoing edges from state α ′ with the eight paths of length 3 that start at that state and, similarly, replacing the outgoing edges from each of the states α ′′ and β with the four paths of length 2 that start at the state.…”

“…Letting Σ and S be as in Example 1, the (S, 2)-encoder in Figure 2 is not an (S, 1, 1)-encoder with respect to the partition (1) of Σ, since both outgoing edges from state α ′ (respectively, state α ′′ ) have the same parity. In fact, using Theorem 1(a) below, it was shown in [14] that for the constraint S and for the partition (1), there is no (S t , 2 t−1 , 2 t−1 )-encoder for any positive integer t, namely, a coding ratio of 1 cannot be achieved by any parity-preserving (fixed-length) encoder, for any t.…”

“…In this section, we extract from [10, several basic definitions and properties pertaining to ordinary (namely, fixed-length) graphs and fixed-length encoders. We then quote the main result of [14], which applies, in particular, to paritypreserving fixed-length encoders.…”

“…Most constructions of parity-preserving encoders that were proposed for commercial use were obtained by ad-hoc methods. In [14], we initiated a study of bi-modal encoders (which include parity-preserving encoders as a special case), focusing on fixed-length encoders; we will summarize the concepts that pertain to the fixed-length case, along with the main results of [14], as part of the background that we provide in Section II below. On the other hand, the existing ad-hoc parity-preserving constructions typically have variable length, where the length p of the input block and the length q of the respective codeword may depend on the encoder state, as well as on the input sequence (the coding ratio, p/q, nevertheless, is still fixed).…”

Roy Adler and the lasting impact of his work

“…The story of this development is best told by Roy [4] and is a wonderful example of the value of unexpected practical applications of pure mathematics. While codes developed by the ACH algorithm are not used much today in actual recording systems, the algorithm continues to be used as a proof of concept in new schemes involving constrained coding, such as codes for radio frequency identification [20], weakly constrained codes [26], and parity preserving codes [40].…”