Concatenated ????- Direct codes and their applications

Let [Formula: see text] denote the finite field with [Formula: see text] elements. While an LCD code [Formula: see text] satisfies the duality criteria [Formula: see text] with [Formula: see text] being the dual code with respect to [Formula: see text], each constituent code [Formula: see text] of a [Formula: see text]-ary [Formula: see text]-Direct code [Formula: see text] is an LCD code in the sense that [Formula: see text] is dual to [Formula: see text] with respect to [Formula: see text], [Formula: see text]. In this paper, a concatenation procedure for the class of [Formula: see text]-ary [Formula: see text]-Direct codes is presented to obtain codes of larger lengths. Further, as the constituent codes of [Formula: see text]-ary [Formula: see text]-Direct codes are known to provide an optimal coding solution for the two-user binary adder channel, the constructed concatenated [Formula: see text]-Direct codes are employed over the [Formula: see text]-user [Formula: see text]-adder channel to harness the usefulness of [Formula: see text]-Direct codes in multi-user coding. The coding and decoding procedures for the constructed class of concatenated codes over the noiseless as well as noisy [Formula: see text]-user [Formula: see text]-adder channel are also presented. The inclusion [Formula: see text] enables and facilitates the error-correction capability required for error correction of [Formula: see text]-Direct codes. With suitable parameters of concatenated [Formula: see text]-Direct codes [Formula: see text], the proposed concatenation and coding procedures are illustrated through examples.