Some Good Codes

Abstract

Let H _n be a Hadamard matrix of order n (see (1.3.5)). In H _n and −H _n we replace −1 by 0. In this way we find 2n rows which are words in F ⁿ₂ . Since any two rows of a Hadamard matrix differ in half of the positions we have constructed an (n, 2n, 1/2n) code. For n = 8 this is an extended Hamming code. For n =32 the code is the one used by Mariner 1969 which was mentioned in Section 2.1. In general these codes are called Hadamard codes.