The average weight enumerator and the maximum likelihood performance of product codes

Product codes are widely used in data-storage, optical and wireless applications. Performance analysis of the product codes in the literature relied on the truncated union lower bounds due to lack of the knowledge of the weight enumerator of most product codes. In this paper, average weight enumerators of arbitrary product codes over F_q are derived. The split weight enumerators of some linear codes are studied and are used in the analysis. The average binary weight enumerator of product Reed Solomon codes is derived. The relationship between the codeword weight and the coordinate weight is discussed for product codes. Consequently, tight bounds on the performance of product codes at low signal to noise ratios are shown.