Circulant Hadamard matrices as HFP-codes of type $C_{4n}\times C_2$

TL;DR

This paper demonstrates that circulant Hadamard codes of length 4n can be represented as HFP-codes of a specific type, and it analyzes their algebraic properties such as rank and kernel dimension.

Contribution

It establishes a new representation of circulant Hadamard codes as HFP-codes of a particular algebraic type and computes their rank and kernel dimension.

Findings

Circulant Hadamard codes are equivalent to HFP-codes of type C_{4n}×C_2.

The rank and kernel dimension of these codes are explicitly computed.

These codes can also be viewed as cocyclic Hadamard codes.

Abstract

We prove that a circulant Hadamard code of length $4n$ can always be seen as an HFP-code (Hadamard full propelinear code) of type $C_{4n}\times C_2$, where $C_2=\langle u\rangle$ or the same, as a cocyclic Hadamard code. We compute the rank and dimension of the kernel of these kind of codes.