A family of codes of length

over GF(

), with

are presented which are constructed by superimposing concatenated codes on a concatenated code. The rate

and the distance ratio

of the new codes satisfy the relation

for sufficiently large values of

and

. The new codes are superior to the comparable Bose-Chaudhuri-Hocquenghem (BCH) codes, for

, in the sense that they contain more codewords. An asymptotically good code constructed using these new codes has a distance ratio greater than those of other asymptotically good codes known to the authors for rates smaller than 0.007.