Formal weight enumerators and Chebyshev polynomials
- PDF / 1,756,181 Bytes
- 18 Pages / 439.37 x 666.142 pts Page_size
- 90 Downloads / 232 Views
Formal weight enumerators and Chebyshev polynomials Masakazu Yamagishi1 Received: 2 March 2020 / Accepted: 28 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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, 𝜀, q) is given in terms of Chebyshev polynomials. According to the parity of n and the sign 𝜀 , 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 𝜀 = −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. Keywords Formal weight enumerator · Chebyshev polynomial · Dickson polynomial · binomial moment Mathematics Subject Classification 11T71 · 12E10
1 Introduction Let q > 0, q ≠ 1 and
* Masakazu Yamagishi [email protected] 1
Department of Mathematics, Nagoya Institute of Technology, Gokiso‑cho, Showa‑ku, Nagoya, Aichi 466‑8555, Japan
13
Vol.:(0123456789)
M. Yamagishi
1 𝜎=√ q
�
1 1
� q−1 , −1
� 𝜏=
1 0
� 0 . −1
(1)
For each n ≥ 1 and 𝜀 ∈ {±1} , define P(n, 𝜀; q) to be the set of homogeneous polynomials 𝜑 ∈ ℂ[x, y] of degree n satisfying
𝜑𝜎 = 𝜀𝜑, 𝜑𝜏 = 𝜑, 𝜑(x, 0) = xn , where the action of
( g=
a c
b d
(2)
) ∈ GL2 (ℂ)
on 𝜑 is define by
𝜑g (x, y) = 𝜑(ax + by, cx + dy).
(3)
For example, it is easy to see that P(1, 𝜀; q) and P(2, −1; q) are empty, while P(2, 1; q) = {W2,q } where
W2,q (x, y) = x2 + (q − 1)y2 . In the case where q is a power of a prime and 𝜑 is the Hamming weight enumerator of a self-dual linear code C of length n over the finite field with q elements, the equality 𝜑𝜎 = 𝜑 holds, and 𝜑𝜏 = 𝜑 if and only if C is divisible by 2, i.e., every codeword of C has even Hamming weight. For example, W2,q is the weight enumerator of a trivial [2, 1, 2] code. According to the Gleason–Pierce theorem [5, 8], the set P(n, 1; q) can be nonempty only when q ∈ {2, 4} or n = 2 , provided that q is a power of a prime. We call a homogeneous polynomial 𝜑 satisfying (2) a formal weight enumerator. The notion of formal weight enumerator was first introduced by Ozeki [6], where, among others, the polynomial
x12 − 33x8 y4 − 33x4 y8 + y12 ∈ P(12, −1;2) plays an important role in connection with modular forms. A systematic investigation of form
Data Loading...
