Set of uniquely decodable codes for overloaded synchronous CDMA

In this paper, we consider the designing of a new set of Uniquely Decodable Codes (UDC) for uncoded synchronous overloaded Code Division Multiple Access (CDMA) for the number of codes exceeding the assigned code length. For the construction, the proposed recursive method at iterationk generates a matrix that can be classified into k orthogonal subsets of different dimensions. Out of them, all besides the largest (binary Hadamard) one are ternary in nature. There resides an inbuilt twin tree structured cross-correlation hierarchy that facilitates an advantageous balance between the auto and intergroup cross-correlation for the signatures in a subset. This opportunity is further leveraged by the proposed multi-stage detector to maintain the uniquely decodable (errorless) nature of the matrices for noiseless transmission. The simple logic of matched filtering serving as the basic designing block of the decoder provides an enormous saving over the complexity of optimum Maximum Likelihood Decoder (MLD). For the noisy channel, we derive the theoretical expression of the average Bit Error Rate (BER) for the individual subset. Also, we explain the role of the two factors (cardinality of the subset, and net level of interference) in being responsible for the non-uniformity in the order of their error performance.