Construction of binary Hadamard codes and their s-PD sets

Abstract

This article presents the construction of binary linear Hadamard codes with parameters (2n,4n,n) over the fields F4 and F8. We generalize this construction to the field $F_{2^{k}}$ to generate Hadamard codes with parameters $\left (2^{k-1}n, 2^{k}n, 2^{k-2}n\right )$ for $k\in \mathbb {N}$ . We construct s-PD sets of size s + 1, a special subset of the permutation automorphism group of a code, which enables the correction of s errors for Hadamard codes over the field F4. We also discuss the decoding algorithm for these Hadamard codes.