Isometry invariant permutation codes and mutually orthogonal Latin squares

Abstract

Commonly, the direct construction and the description of mutually orthogonal Latin squares (MOLS) make use of difference or quasi‐difference matrices. Now there exists a correspondence between MOLS and separable permutation codes. We present separable permutation codes of length 35, 48, 63, and 96 and minimum distance 34, 47, 62, and 95 consisting of 6 goodbreakinfix× 35, 10 goodbreakinfix× 48, 8 goodbreakinfix× 63, and 8 goodbreakinfix× 96 codewords, respectively. Using the correspondence, this gives 6 MOLS for n goodbreakinfix= 35, 10 MOLS for n goodbreakinfix= 48, 8 MOLS for n goodbreakinfix= 63, and 8 MOLS for n goodbreakinfix= 96. The codes are given by generators of an appropriate subgroup U of the isometry group of the symmetric group S n and U‐orbit representatives. This gives an alternative uniform way to describe the MOLS.

A McEliece cryptosystem using permutation codes

“…We can have the private key as a permutation code H (which is a subgroup of S n ) with an efficient decoding algorithm and the public key as the permutation code H = φ(H) for which the decoding algorithm will not work. The isometries in the symmetric group were studied and classified by Farahat [16] who proved that they are of one of the following types:…”

New lower bounds for permutation arrays using contraction

“…Permutation arrays (PAs) with large Hamming distance have been the subject of many recent papers with applications in the design of error correcting codes. New lower bounds for the size of such permutation arrays are given, for example, in [1,2,3,4,5,6,7,12,15,14,19,20,22].…”

A lower bound on permutation codes of distance $n-1$

“…This connection is explored in more detail in [5]. Permutation codes are used in [17] for some recent MOLS constructions.…”