The codes and the lattices of Hadamard matrices

TL;DR

This paper proves the equivalence of extremality conditions for binary, ternary, and Z_4-codes derived from Hadamard matrices, extending results to higher orders and related lattices without relying on classification.

Contribution

It provides two new proofs of the equivalence of extremality of codes and lattices associated with Hadamard matrices, applicable to higher orders and different algebraic structures.

Findings

Proved equivalence of extremality for binary and ternary codes of order 24.

Extended the equivalence to Z_4-codes and lattices for order 48.

Presented proofs that do not depend on the classification of Hadamard matrices.

Abstract

It has been observed by Assmus and Key as a result of the complete classification of Hadamard matrices of order 24, that the extremality of the binary code of a Hadamard matrix H of order 24 is equivalent to the extremality of the ternary code of H^T. In this note, we present two proofs of this fact, neither of which depends on the classification. One is a consequence of a more general result on the minimum weight of the dual of the code of a Hadamard matrix. The other relates the lattices obtained from the binary code and from the ternary code. Both proofs are presented in greater generality to include higher orders. In particular, the latter method is also used to show the equivalence of (i) the extremality of the ternary code, (ii) the extremality of the Z_4-code, and (iii) the extremality of a lattice obtained from a Hadamard matrix of order 48.