Multi-user uniquely decodable (UD) k-ary coding for the multiple-access adder channel is investigated. It is shown that a Tf+g+1-user UD k-ary affine code with code length f+g+1 can be obtained from two Tf-user and Tg-user UD k-ary affine codes. This leads to construct recursively a Tn-user UD k-ary affine code with arbitrary code length n. The total rate of the code tends to be higher than those of all the multi-user UD k-ary codes reported previously as the number of users increases.

Coding scheme for a noisy multiple-access adder channel is proposed. When a T-user δ-decodable affine code C is given a priori, a qT-user λ δ-decodable affine code C* is produced by using a q q matrix B satisfying BA=λ Iq q, e. g. , a Hadamard matrix or a conference matrix. In particular, the case of δ=1 is considered for the practical purposes. A (2n-1)-user uniquely decodable (δ=1) affine code Cn with arbitrary code length n is recursively constructed. When Cn plays a role of C, a q(2n-1)-user λ-decodable affine code C* is obtained. The code length and the number of users of C* are more flexible than those of the Wilson's code. The total rate of the λ-decodable code in this paper tends to be higher than that of the λ-decodable code by Wilson as the number of users increases.

A T-user uniquely decodable (UD) code {C1,C2,,CT} over an integer set {0,1,,k} with arbitrary code length is developed for a multiple-access adder channel (MAAC). Each of the T users is equipped with two codewords, one of which is zero vector. The T-user UD code is used to identify users through the MAAC. It is shown that a T(f+g+1)-user UD code with code length f+g+1 can be arranged from two given T(f)-user and T(g)-user UD codes. This idea makes it possible to construct recursively a T-user UD code for an arbitrary code length n and a positive integer k. The T-user UD code includes the Jevticode.