Algorithmic complexity in coding theory and the minimum distance problem
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Ordered Binary Decision Diagrams and Minimal Trellises
IEEE Transactions on Computers
Constraint complexity of realizations of linear codes on arbitrary graphs
IEEE Transactions on Information Theory
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The trellis complexity s(C) of a block code C is defined as the logarithm of the maximum number of states in the minimal trellis realization of the code. The parameter s(C) governs the complexity of maximum-likelihood decoding, and is considered a fundamental descriptive characteristic of the code in a number of recent works. We derive a new lower bound on s(C) which implies that asymptotically good codes have infinite trellis complexity. More precisely, for i⩾1 let Ci be a code over an alphabet of size q, of length ni, rate Ri, and minimum distance di. The infinite sequence of codes C, C2··· such that ni→∞ when i→∞ is said to be asymptotically good if both Ri and di/ni are bounded away from zero as i→∞. We prove that the complexity s(Ci) increases linearly with ni in any asymptotically good sequence of codes