Codes over Matrix Rings for Space-Time Coded Modulations

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

Skew Cyclic codes over \F_q+u\F_q+v\F_q+uv\F_q

In this paper, we study skew cyclic codes over the ring R=\F_q+u\F_q+v\F_q+uv\F_q, where $u^{2}=u,v^{2}=v,uv=vu$, $q=p^{m}$ and $p$ is an odd prime. We investigate the structural properties of skew cyclic codes over $R$ through a decomposition theorem. Furthermore, we give a formula for the number of skew cyclic codes of length $n$ over $R.