On the Pless-construction and ML decoding of the (48, 24, 12) quadratic residue code

We present a method for maximum likelihood decoding of the (48, 24, 12) quadratic residue code. This method is based on projecting the code onto a subcode with an acyclic Tanner graph, and representing the set of coset leaders by a trellis diagram. This results in a two level coset decoding which can be considered a systematic generalization of the Wagner rule. We show that unlike the (24, 12, 8) Golay code, the (48, 24, 12) code does not have a Pless-construction which has been an open question in the literature. It is determined that the highest minimum distance of a (48, 24) binary code having a Pless-construction is 10, and up to equivalence there are three such codes. Index Terms--Linear block codes, maximum-likelihood decoding, trellis diagram.