Abstract
A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters \((n,\varepsilon ,q)\) is given in terms of Chebyshev polynomials. According to the parity of n and the sign \(\varepsilon\), the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or \(\varepsilon =-1\) using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture.
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Acknowledgements
The author would like to thank Professor Koji Chinen for useful discussions. This work was supported by JSPS KAKENHI Grant No. JP17K05168.
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Yamagishi, M. Formal weight enumerators and Chebyshev polynomials. AAECC 33, 551–568 (2022). https://doi.org/10.1007/s00200-020-00469-1
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DOI: https://doi.org/10.1007/s00200-020-00469-1