AI 中文总结
该研究通过PG(2,q)中二次曲线扩展点集的几何方法，构造了奇素数幂q下长度为q+6和q+7的最优局部性为2的NMDS码，明确其重量分布并证明与现有同类码的等价性差异。
AI 中文摘要
有限域GF(q)上维数为3、长度为n的近最大距离可分（NMDS）码等价于PG(2,q)中的(n,3)-弧。对于每个奇素数幂q，我们通过向PG(2,q)的一条二次曲线添加5个合适的点，构造出一族[q+6,3,q+3] NMDS码，并完整确定其重量分布；存在3种不同的重量枚举式，由参数的两个显式二次特征条件决定。对于每个奇素数幂q，我们构造的码与Fan、Wang和Xu近期提出的[q+6,3,q+3] NMDS码族不存在单项式等价关系，即使两族的重量枚举式重合：区分不变量是与底层弧关联的一组几何数据。将该构型在一条特殊外线上再添加1个点，我们进一步为每个满足q≥11的奇素数幂q构造出[q+7,3,q+4] NMDS码及其重量分布；q≤9时不存在可接受参数；存在性证明结合了精确特征和估计、Weil型特征和估计以及有限计算机验证。我们的证明完全基于几何，且依赖于对二次曲线扩展得到的点集的三割线的简单计数恒等式。所有构造的码都是局部可恢复码，其局部性为2，且是最优的。
英文摘要
Near maximum distance separable (NMDS) codes of dimension 3 and length n over the finite field with q elements are equivalent to (n,3)-arcs in PG(2,q). For every odd prime power q we construct, by adding five suitable points to a conic of PG(2,q), a family of [q+6,3,q+3] NMDS codes and determine their weight distributions completely; three distinct weight enumerators occur, governed by two explicit quadratic-character conditions on the parameters. For every odd prime power q, no code of our family is monomially equivalent to a code of the recent [q+6,3,q+3] NMDS family of Fan, Wang and Xu, even where the weight enumerators of the two families coincide: the separating invariant is a triple of geometric data attached to the underlying arc. Extending the configuration by a sixth point on a distinguished external line, we further obtain [q+7,3,q+4] NMDS codes, together with their weight distributions, for every odd prime power q >= 11 (admissible parameters exist for no q <= 9); the existence proof combines exact and Weil-type character sum estimates with a finite computer verification. Our proofs are purely geometric and rest on a simple counting identity for the trisecant lines of a point set obtained by extending a conic. All the codes constructed are optimal locally recoverable codes with locality 2.
Comments18 pages, 2 figures
