A new table of permutation codes

Permutation codes (or permutation arrays) have received considerable interest in recent years, partly motivated by a potential application to powerline communication. Powerline communication is the transmission of data over the electricity distribution system. This environment is rather hostile to communication and the requirements are such that permutation codes may be suitable. The problem addressed in this study is the construction of permutation codes with a specified length and minimum Hamming distance, and with as many codewords (permutations) as possible. A number of techniques are used including construction by automorphism group and several variations of clique search based on vertex degrees. Many significant improvements are obtained to the size of the best known code

Permutation codes with specified packing radius

Most papers on permutation codes have concentrated on the minimum Hamming distance of the code. An (n, d) permutation code (or permutation array) is simply a set of permutations on n elements in which the Hamming distance between any pair of distinct permutations (or codewords) is at least d. An (n, 2e + 1) or (n, 2e +2) permutation code is able to correct up to e errors. These codes have a potential application to powerline communications. It is known that in an (n, 2e) permutation code the balls of radius e surrounding the codewords may all be pairwise disjoint, but usually some overlap. Thus an (n, 2e) permutation code is generally unable to correct e errors using nearest neighbour decoding. On the other hand, if the packing radius of the code is defined as the largest value of e for which the balls of radius e are all pairwise disjoint, a permutation code with packing radius e can be denoted by [n, e]. Such a permutation code can always correct e errors. In this paper it is shown that, in almost all cases considered, the number of codewords in the best [n, e] code found is substantially greater than the largest number of codewords in the best known (n, 2e + 1) code. Thus the packing radius more accurately specifies the requirement for an e-error-correcting permutation code than does the minimum Hamming distance. The techniques used include construction by automorphism group and several variations of clique search They are enhanced by two theoretical results which make the computations feasibl

A Tutorial on Clique Problems in Communications and Signal Processing

Since its first use by Euler on the problem of the seven bridges of K\"onigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory problems making a large body of the literature readily available for solving and characterizing the complexity of these problems. This tutorial presents a framework for utilizing a particular graph theory problem, known as the clique problem, for solving communications and signal processing problems. In particular, the paper aims to illustrate the structural properties of integer programs that can be formulated as clique problems through multiple examples in communications and signal processing. To that end, the first part of the tutorial provides various optimal and heuristic solutions for the maximum clique, maximum weight clique, and $k$-clique problems. The tutorial, further, illustrates the use of the clique formulation through numerous contemporary examples in communications and signal processing, mainly in maximum access for non-orthogonal multiple access networks, throughput maximization using index and instantly decodable network coding, collision-free radio frequency identification networks, and resource allocation in cloud-radio access networks. Finally, the tutorial sheds light on the recent advances of such applications, and provides technical insights on ways of dealing with mixed discrete-continuous optimization problems

New Upper Bounds for the Size of Permutation Codes via Linear Programming

An (n,d)-permutation code of size s is a subset C of Sn with s elements such that the Hamming distance dH between any two distinct elements of C is at least equal to d. In this paper, we give new upper bounds for the maximal size µ(n,d) of an (n,d)-permutation code of degree n with 11 � n � 14. In order to obtain these bounds, we use the structure of association scheme of the permutation group Sn and the irreducible characters of Sn. The upper bounds for µ(n,d) are determined solving an optimization problem with linear inequalities