Type IV codes over a non-unital ring
Adel Alahmadi, Alaa Altassan, Widyan Basaffar, Hatoon Shoaib, Alexis Bonnecaze, Patrick Solé
Abstract
There is a special local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by [Formula: see text] We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over [Formula: see text] and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.
Topics & Concepts
MathematicsHamming codeLinear codeCommutative ringHamming weightDiscrete mathematicsBlock codeCommutative propertyHamming distanceCombinatoricsAlgorithmDecoding methodsCoding theory and cryptographygraph theory and CDMA systemsError Correcting Code Techniques