Formal weight enumerators and Chebyshev polynomials

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.