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We study product codes based on extended Hamming codes. We focus on their performance at low error rates, which are important for wireless multimedia applications. We present the basis and a complete set of techniques which allows one to analytically evaluate this performance without resorting to extremely long simulations. We present new theoretical results concerning the popular approximation where the bit error rate is nearly equal to the frame error rate times the ratio of the minimum distance to the codeword length. We prove that: 1) binary codes with a transitive automorphism group satisfy this approximation with equality; and 2) extended Hamming product codes belong to this class. Closed-form expressions for their dominant multiplicity values are derived. Analytical curves are plotted, discussed, and validated by comparison with iterative decoding. This analytical approach is then extended to both shortened and punctured codes, which are important for practical design. The first case is solved by applying the extended MacWilliams identity to the dual codes. For punctured codes, we present a new analytical approach for estimating their average performance using a "random" puncturer.