Construction of divisible formal weight enumerators and extremal polynomials not satisfying the Riemann hypothesis

The formal weight enumerators were first introduced by M. Ozeki. They form a ring of invariant polynomials which is similar to that of the weight enumerators of Type II codes. Later, the zeta functions for linear codes were discovered and their theory was developed by I. Duursma. It was generalized...

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Bibliographic Details
Published inarXiv.org
Main Author Chinen, Koji
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 21.05.2018
Subjects
Algorithms
Hypotheses
Invariants
Mathematical analysis
Polynomials
Rings (mathematics)
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Summary:The formal weight enumerators were first introduced by M. Ozeki. They form a ring of invariant polynomials which is similar to that of the weight enumerators of Type II codes. Later, the zeta functions for linear codes were discovered and their theory was developed by I. Duursma. It was generalized to certain invariant polynomials including Ozeki's formal weight enumerators by the present author. One of the famous and important problems is whether extremal weight enumerators satisfy the Riemann hypothesis. In this paper, first we formulate the notion of divisible formal weight enumerators and propose an algorithm for the efficient search of the formal weight enumerators divisible by two. The main tools are the binomial moments. It leads to the discovery of several new families of formal weight enumerators. Then, as a result, we find examples of extremal formal weight enumerators which do not satisfy the Riemann hypothesis.
ISSN:2331-8422