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
