Moment balancing templates for (d, k) - constrained codes and run-length limited sequences

The first-order moment of (d, k) -constrained codes is investigated in this paper. A generalized moment balancing template is proposed to encode a (d, k) sequence into a single insertion of deletion correcting codeword without losing the constraint property. By relocating 0's in moment balancing runs, which appear in a pairwise manner of a (d, k) sequence, the first-order moment of this sequence can be modified to satisfy the Varshamov-Tenengolts construction. With a reasonably large base in the modulo system introduced by the Varshamov-Tenengolts construction, this generalized moment balancing template can be applied to run-lenght limited sequences. The asymptotic bound of the redundancy introduced by the template for (d, k) sequences is of the same order as the universal template for random sequences and, therefore, the redundancy is small and suitable for long sequences of practical interest.http://ieeexplore.ieee.org/servlet/opac?punumber=18nf201

Non-asymptotic Upper Bounds for Deletion Correcting Codes

Explicit non-asymptotic upper bounds on the sizes of multiple-deletion correcting codes are presented. In particular, the largest single-deletion correcting code for $q$-ary alphabet and string length $n$ is shown to be of size at most $\frac{q^n-q}{(q-1)(n-1)}$. An improved bound on the asymptotic rate function is obtained as a corollary. Upper bounds are also derived on sizes of codes for a constrained source that does not necessarily comprise of all strings of a particular length, and this idea is demonstrated by application to sets of run-length limited strings. The problem of finding the largest deletion correcting code is modeled as a matching problem on a hypergraph. This problem is formulated as an integer linear program. The upper bound is obtained by the construction of a feasible point for the dual of the linear programming relaxation of this integer linear program. The non-asymptotic bounds derived imply the known asymptotic bounds of Levenshtein and Tenengolts and improve on known non-asymptotic bounds. Numerical results support the conjecture that in the binary case, the Varshamov-Tenengolts codes are the largest single-deletion correcting codes.Comment: 18 pages, 4 figure