  Reed-Solomon (RS) codes over GF$(2^m)$ have traditionally been the most
popular non-binary codes in almost all practical applications. The distance
properties of RS codes result in excellent performance under hard-decision
bounded-distance decoding. However, efficient and implementable soft decoding
for high-rate (about 0.9) RS codes over large fields (GF(256), say) continues
to remain a subject of research with a promise of further coding gains. In this
work, our objective is to propose and investigate $2^m$-ary codes with
non-trivial binary trace codes as an alternative to RS codes. We derive bounds
on the rate of a $2^m$-ary code with a non-trivial binary trace code. Then we
construct certain subcodes of RS codes over GF($2^m$) that have a non-trivial
binary trace with distances and rates meeting the derived bounds. The
properties of these subcodes are studied and low-complexity hard-decision and
soft-decision decoders are proposed. The decoders are analyzed, and their
performance is compared with that of comparable RS codes. Our results suggest
that these subcodes of RS codes could be viable alternatives for RS codes in
applications.
