发表机构
Tel-Hai University of Kiryat Shmona in the Galilee; MIGAL–Galilee Research Institute, Kiryat Shmona, Israel（加利利基里亚特什莫纳泰尔海大学; 以色列基里亚特什莫纳米格尔-加利利研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对完全图上的二元多节点删除纠正码，扩展循环构造并推导相关度量球公式，得到特定参数下的最优码及对偶性结果。
AI 中文摘要
我们研究坐标为完全无向图的普通边和自环的线性码；节点删除会移除与故障顶点关联的所有坐标。构造结果为二元码。对于三节点删除，我们扩展了已发表的循环构造，允许合适的循环校验斜率依赖于素图长度。显式行列式测试无条件证明，三种固定斜率选择中的一种在无穷多素长度下有效，且给出冗余度为3n-2，比图Singleton界高1比特。我们还给出n=6、8、10、12时的Singleton最优三节点码，以及分离剩余环补全问题的通用普通边框架。当2是奇素数n的原根时，二元多斜率构造可纠正2≤ρ<n时的每一个ρ节点删除，在2≤ρ≤(n+1)/2范围内冗余度为ρn-(ρ-1)。回到任意素幂，我们推导了节点度量球体积的精确生成变换和容斥公式、固定半径渐近式，以及打包、存在和覆盖界。最后，针对互补团删除度量，我们得到精确重量枚举器和Singleton最优节点-团对偶性。
英文摘要
We study linear codes whose coordinates are the ordinary edges and self-loops of complete undirected graphs; a node erasure removes all coordinates incident with a failed vertex. The construction results are binary. For triple-node erasures, we extend the published cyclic construction by allowing a suitable cyclic check slope to depend on the prime graph length. An explicit determinant test proves that one of three fixed slope choices works at infinitely many prime lengths, unconditionally, and gives redundancy $3n-2$, one bit above the graph Singleton bound. We also give Singleton-optimal triple-node codes at $n=6,8,10,12$, together with a general ordinary-edge framework that isolates the remaining loop-completion problem. When $2$ is primitive modulo an odd prime $n$, a binary multi-slope construction corrects every $ρ$-node erasure for $2\leqρ<n$, with redundancy $ρn-(ρ-1)$ in the range $2\leqρ\leq(n+1)/2$. Returning to arbitrary prime powers, we derive exact generating transforms and inclusion--exclusion formulas for node-metric ball volumes, fixed-radius asymptotics, and packing, existence, and covering bounds. Finally, for the complementary clique-erasure metric, we obtain an exact weight enumerator and a Singleton-optimal node--clique duality.
Comments48 pages. Revised exposition and layout; clarified the decoder analysis and the scope of supporting statements
