Spectral Shape of Doubly-Generalized LDPC Codes: Efficient and Exact Evaluation
This paper analyzes the asymptotic exponent of the weight spectrum for irregular Doubly-Generalized LDPC (DGLDPC) codes. In the process, an efficient numerical technique for its evaluation is presented, involving the solution of a 4 ? 4 system of polynomial equations. The expression is consistent with previous results, including the case where the normalized weight or stopping set size tends to zero. The spectral shape is shown to admit a particularly simple form in the special case where all variable nodes are repetition codes of the same degree, a case which includes Tanner codes; for this case it is also shown how certain symmetry properties of the local weight distribution at the CNs induce a symmetry in the overall weight spectral shape function.