一类扭曲广义Reed-Solomon码
信息论
2022-04-01 v2 math.IT
摘要
设F q \mathbb{F}_q F q q q q F q ∗ \mathbb{F}_q^* F q ∗ F q \mathbb{F}_q F q C ℓ ( D , k , η , v ⃗ ) ⊂ F q n C_\ell(D, k, \eta, \vec{v})\subset \mathbb{F}_q^n C ℓ ( D , k , η , v ) ⊂ F q n ( v 1 v 2 ⋯ v n v 1 α 1 v 2 α 2 ⋯ v n α n ⋮ ⋮ ⋱ ⋮ v 1 α 1 ℓ − 1 v 2 α 2 ℓ − 1 ⋯ v n α n ℓ − 1 v 1 α 1 ℓ + 1 v 2 α 2 ℓ + 1 ⋯ v n α n ℓ + 1 ⋮ ⋮ ⋱ ⋮ v 1 α 1 k − 1 v 2 α 2 k − 1 ⋯ v n α n k − 1 v 1 ( α 1 ℓ + η α 1 q − 2 ) v 2 ( α 2 ℓ + η α 2 q − 2 ) ⋯ v n ( α n ℓ + η α n q − 2 ) ) \left(\begin{array}{cccc} v_{1} & v_{2} & \cdots & v_{n} \ v_{1} \alpha_{1} & v_{2} \alpha_{2} & \cdots & v_{n} \alpha_{n} \ \vdots & \vdots & \ddots & \vdots \ v_{1} \alpha_{1}^{\ell-1} & v_{2} \alpha_{2}^{\ell-1} & \cdots & v_{n} \alpha_{n}^{\ell-1} \ v_{1} \alpha_{1}^{\ell+1} & v_{2} \alpha_{2}^{\ell+1} & \cdots & v_{n} \alpha_{n}^{\ell+1} \ \vdots & \vdots & \ddots & \vdots \ v_{1} \alpha_{1}^{k-1} & v_{2} \alpha_{2}^{k-1} & \cdots & v_{n} \alpha_{n}^{k-1} \ v_{1}\left(\alpha_{1}^{\ell}+\eta\alpha_{1}^{q-{2}}\right) & v_{2}\left(\alpha_{2}^{\ell}+ \eta \alpha_{2}^{q-2}\right) &\cdots & v_{n}\left(\alpha_{n}^{\ell}+\eta\alpha_{n}^{q-2}\right) \end{array}\right) ( v 1 v 2 ⋯ v n v 1 α 1 v 2 α 2 ⋯ v n α n ⋮ ⋮ ⋱ ⋮ v 1 α 1 ℓ − 1 v 2 α 2 ℓ − 1 ⋯ v n α n ℓ − 1 v 1 α 1 ℓ + 1 v 2 α 2 ℓ + 1 ⋯ v n α n ℓ + 1 ⋮ ⋮ ⋱ ⋮ v 1 α 1 k − 1 v 2 α 2 k − 1 ⋯ v n α n k − 1 v 1 ( α 1 ℓ + η α 1 q − 2 ) v 2 ( α 2 ℓ + η α 2 q − 2 ) ⋯ v n ( α n ℓ + η α n q − 2 ) ) 0 ≤ ℓ ≤ k − 1 0\leq \ell\leq k-1 0 ≤ ℓ ≤ k − 1 D = { α 1 , α 2 , ⋯ , α n } ⊆ F q ∗ D=\{\alpha_{1},\alpha_{2},\cdots, \alpha_{n}\}\subseteq \mathbb{F}_q^* D = { α 1 , α 2 , ⋯ , α n } ⊆ F q ∗ v ⃗ = ( v 1 , v 2 , ⋯ , v n ) ∈ ( F q ∗ ) n \vec{v}=(v_1,v_2,\cdots,v_n)\in (\mathbb{F}_q^*)^n v = ( v 1 , v 2 , ⋯ , v n ) ∈ ( F q ∗ ) n η ∈ F q ∗ \eta\in\mathbb{F}_q^* η ∈ F q ∗ C ℓ ( D , k , η , v ⃗ ) C_\ell(D, k, \eta, \vec{v}) C ℓ ( D , k , η , v ) ℓ = k − 1 \ell=k-1 ℓ = k − 1 C k − 1 ( D , k , η , v ⃗ ) C_{k-1}(D, k, \eta, \vec{v}) C k − 1 ( D , k , η , v )
引用
@article{arxiv.2202.09011,
title = {A class of twisted generalized Reed-Solomon codes},
author = {Jun Zhang and Zhengchun Zhou and Chunming Tang},
journal= {arXiv preprint arXiv:2202.09011},
year = {2022}
}
备注
11 pages