A new table of permutation codes

Abstract

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 codes.

Constructing permutation arrays from groups

“…On the other hand, the coset method starting with the group G = AGL(1, 16) found 5, 739 cosets of G for Hamming distance 9 and, hence, obtained a lower bound of 1, 377, 360 for M (16, 9). Also, in [21] the authors computed a PA of size 20, 908, 800 for M(12, 4) and then stated "... it is too large to check fully, but has been extensively checked." In contrast, the coset method starting with the Mathieu group G of order 12 [4,7,8] found 638 cosets of G for Hamming distance 4 and, hence, obtained a lower bound of 60, 635, 520 for M (12, 4).…”

A branch and bound approach to permutation codes

“…As reported in [12] the optimal solution of this problem is M (6, 5) = 18 and one of the many possible optimal codes is …”

“…As well as white Gaussian noise the codes must combat permanent narrow band noise from electrical equipment or magnetic fields, and impulse noise. Most of the approaches presented in the literature are based on linear programming [7], [8] or on group theory ideas [7], [9], [10], [11], which have more recently been amalgamated with optimization, mainly based on search techniques [12], [13], [14].…”

Graph Colouring and Branch and Bound Approaches for Permutation Code Algorithms

“…Thus if M(n, d) denotes the maximum number of codewords in an (n, d) permutation code it is important to determine M(n, d). The most complete contributions to lower bounds for M(n, d) can be found in [6,20]. Some studies considering upper bounds have appeared in [5,10,[21][22][23]26], while in [24,25] studies on the structure of optimal codes have been presented.…”