BINARY LINEAR CODES WITH NEAR-EXTREMAL MAXIMUM DISTANCE

Let C denote a binary linear code with length n all of whose coordinates are essential; i.e., for each coordinate there is a codeword that is not zero in that position. Then the maximum distance D is strictly bigger than n/2, and the extremum D = (n + 1)/2 is attained exactly by punctured Hadamard codes. In this paper, we classify binary linear codes with D = n/2 + 1. All of these codes can be produced from punctured Hadamard codes in one of essentially three different ways, each having a transparent description.