Quadratic Forms and Space-Time Block Codes from Generalized Quaternion and Biquaternion Algebras

TL;DR

This paper explores the use of generalized quaternion and biquaternion algebras over various fields to construct new space-time block codes, providing algebraic criteria and explicit examples for division algebras used in coding.

Contribution

It introduces quadratic form criteria for identifying division algebras and constructs new families of space-time block codes from these algebraic structures.

Findings

Constructed explicit infinite families of (bi-)quaternion division algebras.

Developed quadratic form criteria using Springer's theorem.

Produced new 2x2 and 4x4 space-time block codes.

Abstract

In the context of space-time block codes (STBCs), the theory of generalized quaternion and biquaternion algebras (i.e., tensor products of two quaternion algebras) over arbitrary base fields is presented, as well as quadratic form theoretic criteria to check if such algebras are division algebras. For base fields relevant to STBCs, these criteria are exploited, via Springer's theorem, to construct several explicit infinite families of (bi-)quaternion division algebras. These are used to obtain new $2\x 2$ and $4\x 4$ STBCs.