2022/02/18 by Jun Zhang, Zhengchun Zhou, Zhang, Jun +3 · 8 citations
Computer Science · Social Sciences · #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Computer and information sciences #Information Theory (cs.IT) #Islamic Finance and Communication
paper · pdf · doi:10.48550/arxiv.2202.09011
openalex publication_date 2022/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbbFq be a finite field of size q and \mathbbFq^* the set of non-zero elements of \mathbbFq. In this paper, we study a class of twisted generalized Reed-Solomon code C_ℓ(D, k, η, v)⊂ \mathbbFqn generated by the following matrix (v1 · amp; v2 · amp; ⋯ · amp; vn
v1 α1 · amp; v2 α2 · amp; ⋯ · amp; vn αn
⋮ · amp; ⋮ · amp; \ddots · amp; ⋮
v1 α1ℓ-1 · amp; v2 α2ℓ-1 · amp; ⋯ · amp; vn αnℓ-1
v1 α1ℓ+1 · amp; v2 α2ℓ+1 · amp; ⋯ · amp; vn αnℓ+1
⋮ · amp; ⋮ · amp; \ddots · amp; ⋮
v1 α1k-1 · amp; v2 α2k-1 · amp; ⋯ · amp; vn αnk-1
v1(α1ℓ+ηα1^q-2) · amp; v2(α2ℓ+ ηα2q-2) · amp;⋯ · amp; vn(αnℓ+ηαnq-2)) where 0≤ ℓ≤ k-1, the evaluation set D=\α1,α2,⋯, αn\⊆ \mathbbFq^*, scaling vector v=(v1,v2,⋯,vn)∈ (\mathbbFq^*)n and η∈\mathbbFq^*. The minimum distance and dual code of C_ℓ(D, k, η, v) will be determined. For the special case ℓ=k-1, a sufficient and necessary condition for Ck-1(D, k, η, v) to be self-dual will be given. We will also show that the code is MDS or near-MDS. Moreover, a complete classification when the code is near-MDS or MDS will be presented.