New Bounds for Permutation Codes in Ulam Metric

New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths. Second, several probabilistic lower bounds are developed, which improve on the known lower bounds for large minimum distances. The results of a computer search for permutation codes are also presented.Comment: To be presented at ISIT 2015, 5 page

Permutation Coding with Injections for Modified PAM System

Arriving at a good combination of coding andmodulation schemes that can achieve good error correctionconstitutes a challenge in digital communication systems. In thiswork, we explore the combination of permutation coding (PC)and pulse amplitude modulation (PAM) for mitigating channelerrors in the presence of background noise and jitter. Since PAMis characterised with bi-polar constellations, Euclidean distance isa good choice for predicting the performance of such coded modulationsetup. In order to address certain challenges facing PCs,we therefore introduce injections in the coding system, togetherwith a modified form of PAM system. This modification entailsconstraining the PAM constellations to the size of the codeword’ssymbol. The results obtained demonstrate the strength of themodified coded PAM system over the conventional PC codedPAM system

Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three

The concept of group divisible codes, a generalization of group divisible designs with constant block size, is introduced in this paper. This new class of codes is shown to be useful in recursive constructions for constant-weight and constant-composition codes. Large classes of group divisible codes are constructed which enabled the determination of the sizes of optimal constant-composition codes of weight three (and specified distance), leaving only four cases undetermined. Previously, the sizes of constant-composition codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table

The PBD-Closure of Constant-Composition Codes

We show an interesting PBD-closure result for the set of lengths of constant-composition codes whose distance and size meet certain conditions. A consequence of this PBD-closure result is that the size of optimal constant-composition codes can be determined for infinite families of parameter sets from just a single example of an optimal code. As an application, the size of several infinite families of optimal constant-composition codes are derived. In particular, the problem of determining the size of optimal constant-composition codes having distance four and weight three is solved for all lengths sufficiently large. This problem was previously unresolved for odd lengths, except for lengths seven and eleven.Comment: 8 page