Z4-linear Hadamard and extended perfect codes

TL;DR

This paper classifies and constructs $Z_4$-linear Hadamard and extended perfect codes for lengths greater than 8, providing a recursive method for their construction and counting their equivalence classes.

Contribution

It establishes the exact number of nonequivalent $Z_4$-linear Hadamard and extended perfect codes for certain lengths and introduces a recursive construction method.

Findings

Exactly $[(k-1)/2]$ nonequivalent $Z_4$-linear Hadamard codes for $N=2^k > 8

Exactly $[(k+1)/2]$ nonequivalent $Z_4$-linear extended perfect codes for $N=2^k > 8

A recursive construction method for $Z_4$-linear Hadamard codes

Abstract

If $N=2^k > 8$ then there exist exactly $[(k-1)/2]$ pairwise nonequivalent $Z_4$-linear Hadamard $(N,2N,N/2)$-codes and $[(k+1)/2]$ pairwise nonequivalent $Z_4$-linear extended perfect $(N,2^N/2N,4)$-codes. A recurrent construction of $Z_4$-linear Hadamard codes is given.