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The weight hierarchy of product codes

H.G. Schaathun
2000 IEEE Transactions on Information Theory  
The weight of a code is the number of coordinate positions where no codeword is zero. The th minimum weight is the least weight of an -dimensional subcode. Wei and Yang gave a conjecture about the minimum weights for some product codes. In this correspondence, we will find a relation between product codes and the Segre embedding of a pair of projective systems, and we use this to prove the conjecture. Index Terms-Product code, projective system (projective multiset), Segre embedding, weight
more » ... bedding, weight hierarchy. Abstract-The optimum rate-compatible repetition convolutional (RCRC) codes with different parent encoder rates and different constraint lengths are presented in this correspondence. They are constructed according to the optimum distance spectrum (ODS) criterion. The obtained codes provide, e.g., significant throughput gains compared to simple repetition codes, when applied in hybrid type II automatic repeat request (ARQ) schemes. Index Terms-Optimum distance spectrum, rate-compatible repetition convolutional (RCRC) codes, rate-compatible punctured convolutional (RCPC) codes), unequal error protection.
doi:10.1109/18.887876 fatcat:7bdifbhudvanlabixjndmsheum