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A Riemann hypothesis analogue for invariant rings
AbstractA Riemann hypothesis analogue for coding theory was introduced by I.M. Duursma [A Riemann hypothesis analogue for self-dual codes, in: A. Barg, S. Litsyn (Eds.), Codes and Association Schemes (Piscataway, NJ, 1999), American Mathematical Society, Providence, RI, 2001, pp. 115–124]. In this paper, we extend zeta polynomials for linear codes to ones for invariant rings, and we investigate whether a Riemann hypothesis analogue holds for some concrete invariant rings. Also we shall show that there is some subring of an invariant ring such that the subring is not an invariant ring but extremal polynomials all satisfy the Riemann hypothesis analogue
Results on zeta functions for codes
We give a new and short proof of the Mallows-Sloane upper bound for self-dual
codes. We formulate a version of Greene's theorem for normalized weight
enumerators. We relate normalized rank-generating polynomials to two-variable
zeta functions. And we show that a self-dual code has the Clifford property,
but that the same property does not hold in general for formally self-dual
codes.Comment: 12 page
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