Binary Linear Codes and Symmetric Generation of Finite Simple Groups

In this paper, we study a new combinatorial method to construct decodable binary linear codes for which the automorphism groups are generated by sets of involutory symmetric generators. In this method codewords as elements of a group are represented as permutations in Sn followed by words in the n involutory symmetric generators. Transformation between elements written in symmetric representations and permutations in list forms, with the well-known Hamming distance, is given. Although it is feasible to handle permutations of reasonably large sizes and perform composition operations, transmitting and recording such elements is inconvenient. Symmetric representations of the elements of a code automorphism group have an advantage over permutation representations in terms of conciseness as well as ease of conducting operations. Mathematics Subject Classification: 20B20, 94B05, 94B25 Keyword: Coding theory, errors detecting-correcting, permutation groups, symmetric generation of groups