2024/01/26 by Dipak K. Bhunia, Bhunia, Dipak K., Cristina Fernández-Córdoba +3
Computer Science · Engineering · #Coding theory and cryptography #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2401.14799
openalex publication_date 2024/01/26 · openalex created_date 2024/01/30 · openalex updated_date 2026/08/01
The ℤ2ℤ4ℤ8-additive codes are subgroups of ℤ2α1 × ℤ4α2 × ℤ8α3. A ℤ2ℤ4ℤ8-linear Hadamard code is a Hadamard code which is the Gray map image of a ℤ2ℤ4ℤ8-additive code. A recursive construction of ℤ2ℤ4ℤ8-additive Hadamard codes of type (α1,α2, α3;t1,t2, t3) with α1 ≠ 0, α2 ≠ 0, α3 ≠ 0, t1≥ 1, t2 ≥ 0, and t3≥ 1 is known. In this paper, we generalize some known results for ℤ2ℤ4-linear Hadamard codes to ℤ2ℤ4ℤ8-linear Hadamard codes with α1 ≠ 0, α2 ≠ 0, and α3 ≠ 0. First, we show for which types the corresponding ℤ2ℤ4ℤ8-linear Hadamard codes of length 2t are nonlinear. For these codes, we compute the kernel and its dimension, which allows us to give a partial classification of these codes. Moreover, for 3 ≤ t ≤ 11, we give a complete classification by providing the exact amount of nonequivalent such codes. We also prove the existence of several families of infinite such nonlinear ℤ2ℤ4ℤ8-linear Hadamard codes, which are not equivalent to any other constructed ℤ2ℤ4ℤ8-linear Hadamard code, nor to any ℤ2ℤ4-linear Hadamard code, nor to any previously constructed ℤ2s-linear Hadamard code with s≥ 2, with the same length 2t.