Asian Journal of Mathematics and Computer Research

Shi et al. (2021) and Cheng (2024) constructed LCD double Toeplitz codes from tridiagonal symmetric and skew-symmetric Toeplitz matrices by factorizing Dickson polynomials. This paper extends their work to linear codes generated by anti-tridiagonal Hankel matrices with zero on the sub-diagonal, denoted H_2n+1(b, 0, c). Using a permutation involution, it is shown that H_2n+1(b, 0, c) is similar to a symmetric tridiagonal 2-Toeplitz matrix with zero main diagonal, allowing the factorization of the Hankel matrix’s characteristic polynomial via Dickson polynomials. Based on this factorization, necessary and sufficient conditions for the code C_2n+1(b, 0, c) = [I_2n+1,H_2n+1(b, 0, c)] to be LCD are derived over finite fields of both even and odd characteristic. For even characteristic, the LCD condition is expressed in terms of roots of unity and parameters b, c, including an extension when gcd(p, n+1) ̸= 1. For odd characteristic, conditions involve primitive 2(n+1)-th roots of unity and the element μ with μ² = −1. The results are further generalized to the case p^r ∥ (n + 1) by reducing to primitive (m + 1)-th or 2(m + 1)-th roots. These constructions provide new families of LCD codes from Hankel matrices, complementing existing Toeplitz-based constructions and offering flexible parameter choices for side-channel and fault-injection attack resistant cryptography. An example over F4 is given.