The necessary and sufficient condition of the channel capacity is rigorously formulated for the$N$-user discrete memoryless multiple-access channel (MAC). The essence is to invoke anelementaryMAC where sizes of input alphabets are not greater than the size of output alphabet. The main objective is to demonstrate that the channel capacity of an MAC is achieved by an elementary MAC included in the original MAC. The proof is quite straightforward by the very definition of the elementary MAC. The second objective is to prove that the Kuhn–Tucker conditions of the elementary MAC are sufficient (obviously necessary) for the channel capacity. The latter proof requires two distinctive properties of the MAC: Every solution of the Kuhn–Tucker conditions is a local maximum on the domain of all possible input probability distributions (IPDs), and then particularly for the elementary MAC a set of IPDs for which the value of the mutual information is not smaller than the arbitrary positive number is connected on the domain. As a result, in respect of the channel capacity, the MAC in general can be regarded as an aggregate of a finite number of elementary MACs.

Error-correcting non-binary signature code is proposed. A 2j-1-decodable (k + 1)-ary signature code with code length 2j- 1 is recursively constructed. The code is used to identify users through a multiple-access adder channel, even in the presence of channel noise.

User identification by signature code is considered for noisy multiple-access adder channel. An n/2-decodable signature code is developed from a Hadamard matrix of order n. The code is used to identify users through the multiple-access adder channel even in the presence of channel noise. A decoding rule is provided to correct lfloor(n/2 - 1)/2rfloor errors and then to identify users

The necessary and sufficient condition for constructing a spreading set with decodability is investigated. It is proved that for a given ${\delta}$ -decodable spreading set and a $q \times q$ square matrix $H$ with components $1$ or $-1$ , a ${q\delta}$ -decodable spreading set $S^\ast$ is obtained if and only if $H$ is a Hadamard matrix. In addition, a decoding rule with error correction and message data detection is provided.

Error-correcting signature code is proposed. An n/2-decodable signature code with code length n - 1 and cardinality n - 1 is developed from an Hadamard matrix of order n. The code is used to identify users through the multiple-access adder channel, even in the presence of channel noise

This paper investigates the channel configuration of multiple-access channel (MAC) by decomposing the MAC into finite subMAC and also proves that there exists an input probability that achieves boundary of a capacity region and the total capacity of the original MAC.