Codes Over Matrix Rings for Space-Time Coded Modulations

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, by taking the quotient of the algebra by a two-sided ideal which leads to matrices over finite alphabets for the outer code. In this paper, 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 2×2 Golden code by using an ideal above the prime 2 leads to consider codes over either M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) or M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> [i]), both being noncommutative alphabets. By identifying them as algebras over a finite field or a finite ring respectively, we establish an unexpected connection with classical error-correcting codes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> and F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> [i]. Matrix rings of higher dimension, suitable for 3×3 and 4×4 perfect codes, give rise to more complex examples.