论文标题
一类扭曲的广义芦苇 - 固体代码
A class of twisted generalized Reed-Solomon codes
论文作者
论文摘要
令$ \ mathbb {f} _q $为尺寸$ q $和$ \ mathbb {f} _q^*$的有限字段，$ \ mathbb {f} _q $的非零元素集。在本文中，我们研究了一类扭曲的通用芦苇 - 溶剂代码$ c_ \ ell（d，k，η，\ vec {v}）\ subset \ subset \ mathbb {f} _q^n $由以下矩阵\ [\ left（\ lews） v_ {n} \\ v_ {1}α_{1}＆v_ {2}α_{2}＆\ cdots＆v_ {n}α_{n} n} \ \ \\\\\\\\\\\\\\\\\\\\\\ vdots＆\ vdots＆\ vdots＆\ ddots＆\ ddots＆\ ddots＆\ vdots＆\ v _} v_ {2}α_{2}^{\ ell-1}＆\ cdots＆v_ {n}α_{n}^{\ ell-1} \\ v_ {1}α__{1}α_{1}^{\ ell+1} v_ {n}α_{n}^{\ ell+1} \\ \ \ \ \ vdots＆\ vdots＆\ ddots＆\ ddots＆\ vdots＆\ vdots \\ v_ {1}α__{1}^{1}^{k-1} {k-1}＆v_} α_{n}^{k-1} \\ v_ {1} \ left（α_{1}^{\ ell}+ηα_{1}^{q- {q- {2}}}} \ right） ηα_{2}^{q-2} \ right） $ d = \ {α_{1}，α_{2}，\ cdots，α_{n} \} \ subseteq \ subseteq \ mathbb {f} _q^*$，缩放vector $ \ \ vec $ \ vec {v} （\ mathbb {f} _q^*）^n $和$η\ in \ mathbb {f} _q^*$。将确定$ c_ \ ell（d，k，η，\ vec {v}）$的最小距离和双代码。对于特殊情况$ \ ell = k-1，$是$ c_ {k-1}（d，k，η，\ vec {v}）$的足够且必要的条件。我们还将证明代码是MDS或近MD。此外，当代码为接近MD或MD时，将进行完整的分类。
Let $\mathbb{F}_q$ be a finite field of size $q$ and $\mathbb{F}_q^*$ the set of non-zero elements of $\mathbb{F}_q$. In this paper, we study a class of twisted generalized Reed-Solomon code $C_\ell(D, k, η, \vec{v})\subset \mathbb{F}_q^n$ generated by the following matrix \[ \left(\begin{array}{cccc} v_{1} & v_{2} & \cdots & v_{n} \\ v_{1} α_{1} & v_{2} α_{2} & \cdots & v_{n} α_{n} \\ \vdots & \vdots & \ddots & \vdots \\ v_{1} α_{1}^{\ell-1} & v_{2} α_{2}^{\ell-1} & \cdots & v_{n} α_{n}^{\ell-1} \\ v_{1} α_{1}^{\ell+1} & v_{2} α_{2}^{\ell+1} & \cdots & v_{n} α_{n}^{\ell+1} \\ \vdots & \vdots & \ddots & \vdots \\ v_{1} α_{1}^{k-1} & v_{2} α_{2}^{k-1} & \cdots & v_{n} α_{n}^{k-1} \\ v_{1}\left(α_{1}^{\ell}+ηα_{1}^{q-{2}}\right) & v_{2}\left(α_{2}^{\ell}+ ηα_{2}^{q-2}\right) &\cdots & v_{n}\left(α_{n}^{\ell}+ηα_{n}^{q-2}\right) \end{array}\right) \] where $0\leq \ell\leq k-1,$ the evaluation set $D=\{α_{1},α_{2},\cdots, α_{n}\}\subseteq \mathbb{F}_q^*$, scaling vector $\vec{v}=(v_1,v_2,\cdots,v_n)\in (\mathbb{F}_q^*)^n$ and $η\in\mathbb{F}_q^*$. The minimum distance and dual code of $C_\ell(D, k, η, \vec{v})$ will be determined. For the special case $\ell=k-1,$ a sufficient and necessary condition for $C_{k-1}(D, k, η, \vec{v})$ 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.
