Codes over Matrix Rings for Space-Time Coded Modulations

TL;DR

This paper explores the design of outer codes over matrix rings derived from inner space-time codes for MIMO channels, analyzing metrics like Hamming and Bachoc distances to improve diversity and coding gain.

Contribution

It introduces a novel approach to outer code design using quotient algebras over finite rings, extending the algebraic framework of space-time coding.

Findings

Outer codes can be constructed over matrix rings derived from algebra quotients.

Determinant criteria induce metrics such as Hamming and Bachoc distances on outer codes.

Partitioning of the Golden code over prime ideals leads to codes over non-commutative alphabets.

Abstract

It is known that, for transmission over quasi-static MIMO fading channels with n transmit antennas, diversity can be obtained by using an inner fully diverse space-time block code while coding gain, derived from the determinant criterion, comes from an appropriate outer code. When the inner code has a cyclic algebra structure over a number field, as for perfect space-time codes, an outer code can be designed via coset coding. More precisely, we take the quotient of the algebra by a two-sided ideal which leads to a finite alphabet for the outer code, with a cyclic algebra structure over a finite field or a finite ring. We show that the determinant criterion induces various metrics on the outer code, such as the Hamming and Bachoc distances. When n=2, partitioning the 2x2 Golden code by using an ideal above the prime 2 leads to consider codes over either M2(F_2) or M2(F_2[i]), both being non-commutative alphabets. Matrix rings of higher dimension, suitable for 3x3 and 4x4 perfect codes, give rise to more complex examples.