Counting subset sums of finite abelian groups
Journal of Combinatorial Theory Series A
Stopping set distributions of algebraic geometry codes from elliptic curves
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
Generalised jacobians in cryptography and coding theory
WAIFI'12 Proceedings of the 4th international conference on Arithmetic of Finite Fields
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The minimum distance is one of the most important combinatorial characterizations of a code. The maximum-likelihood decoding problem is one of the most important algorithmic problems of a code. While these problems are known to be hard for general linear codes, the techniques used to prove their hardness often rely on the construction of artificial codes. In general, much less is known about the hardness of the specific classes of natural linear codes. In this correspondence, we show that both problems are NP-hard for algebraic geometry codes. We achieve this by reducing a well-known NP-complete problem to these problems using a randomized algorithm. The family of codes in the reductions is based on elliptic curves. They have positive rates, but the alphabet sizes are exponential in the block lengths.