Self-dual codes and Hadamard matrices (Q1180166)

A further improvement of the method (Bhargava, Stein, Mezzaroba, Salwach) of binary self-orthogonal code construction is given using combinatorial designs and Hadamard matrices. The following theorem is proved: Let \(H\) be a Hadamard matrix of order \(n=8t+4\) with some row (and hence all rows) containing an odd number of 1's. Then the matrix \((IA)\) \((A=(H+J)/2\), \(J\) is the all one matrix) generates a self-dual doubly-even code of length \(2n\). The minimum distance of \(C\) is at last 8 if and only if each row and column of \(H\) contains at least 7 1's. The external doubly-even self- dual codes for \(t=2,3\) are constructed.