A class of twisted generalized Reed-Solomon codes
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Let 𝔽_q be a finite field of size q and 𝔽_q^* the set of non-zero elements of 𝔽_q. In this paper, we study a class of twisted generalized Reed-Solomon code C_ℓ(D, k, η, v⃗)⊂𝔽_q^n generated by the following matrix ([ v_1 v_2 ⋯ v_n; v_1α_1 v_2α_2 ⋯ v_nα_n; ⋮ ⋮ ⋱ ⋮; v_1α_1^ℓ-1 v_2α_2^ℓ-1 ⋯ v_nα_n^ℓ-1; v_1α_1^ℓ+1 v_2α_2^ℓ+1 ⋯ v_nα_n^ℓ+1; ⋮ ⋮ ⋱ ⋮; v_1α_1^k-1 v_2α_2^k-1 ⋯ v_nα_n^k-1; v_1(α_1^ℓ+ηα_1^q-2) v_2(α_2^ℓ+ ηα_2^q-2) ⋯ v_n(α_n^ℓ+ηα_n^q-2) ]) where 0≤ℓ≤ k-1, the evaluation set D={α_1,α_2,⋯, α_n}⊆𝔽_q^*, scaling vector v⃗=(v_1,v_2,⋯,v_n)∈ (𝔽_q^*)^n and η∈𝔽_q^*. The minimum distance and dual code of C_ℓ(D, k, η, v⃗) will be determined. For the special case ℓ=k-1, a sufficient and necessary condition for C_k-1(D, k, η, v⃗) 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.
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