论文标题
从属于$ g \ geq 1 $的功能字段的自动形态群中的本地可回收代码
Locally recoverable codes from automorphism groups of function fields of genus $g \geq 1$
论文作者
论文摘要
本地可恢复的代码是一个代码，使得可以从其他坐标的一小部分的值中恢复任何单个坐标的值。当我们有$δ$的非重叠的基数$ r_i $时，可用于恢复缺失的坐标，我们说的是，线性代码$ \ nathcal {c} $带长度$ n $，dimension $ k $，最小距离$ d $ a $ d $ a $ d $ a $（r_1，\ ldots，r_Δ）$ - r_Δ）$ - loctality and $ - by $ by $ by $ [n，n，d [n，d; r_1，r_2，\ dots，r_Δ]。$在本文中，我们为这些代码的最小距离提供了一个新的上限。使用有限数量的基数子组$ r_i+1 $的自动形态组的函数字段$ \ mathcal {f} | \ Mathbb {f} _q $ of属$ g \ geq 1 $，我们提出了$ [n，k，d; r_1，r_2，\ dots，r_δ] $ - 代码，并将结果应用于一些众所周知的功能场。
A Locally Recoverable Code is a code such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. When we have $δ$ non overlapping subsets of cardinality $r_i$ that can be used to recover the missing coordinate we say that a linear code $\mathcal{C}$ with length $n$, dimension $k$, minimum distance $d$ has $(r_1,\ldots, r_δ)$-locality and denote it by $[n, k, d; r_1, r_2,\dots, r_δ].$ In this paper we provide a new upper bound for the minimum distance of these codes. Working with a finite number of subgroups of cardinality $r_i+1$ of the automorphism group of a function field $\mathcal{F}| \mathbb{F}_q$ of genus $g \geq 1$, we propose a construction of $[n, k, d; r_1, r_2,\dots, r_δ]$-codes and apply the results to some well known families of function fields.
