论文标题
关于最小代码的三种组合观点
Three Combinatorial Perspectives on Minimal Codes
论文作者
论文摘要
我们开发了三种组合风味的方法，以研究有限几何形状中最小代码的结构和切割阻塞集的结构，每种几何都具有特定的应用。第一种方法使用代数组合学中的技术，通过Alon-Füredi定理和组合nullstellensatz在线性代码中描述了支持。第二种方法结合了编码理论和统计数据的方法，以比较最小代码中非零权重的平均值和方差。最后，第三种方法将最小代码视为切割阻塞集，并使用有限几何形状的扩展理论进行研究。我们彼此应用和结合了这些方法，我们在最小代码的参数上得出了几个新的界限和约束。此外，我们获得了有限投影空间中小型基数的切割封锁集的两个新结构。反过来，这些使我们能够对给定字段和维度的最小代码进行明确的构造。
We develop three approaches of combinatorial flavour to study the structure of minimal codes and cutting blocking sets in finite geometry, each of which has a particular application. The first approach uses techniques from algebraic combinatorics, describing the supports in a linear code via the Alon-Füredi Theorem and the Combinatorial Nullstellensatz. The second approach combines methods from coding theory and statistics to compare the mean and variance of the nonzero weights in a minimal code. Finally, the third approach regards minimal codes as cutting blocking sets and studies these using the theory of spreads in finite geometry. Applying and combining these approaches with each other, we derive several new bounds and constraints on the parameters of minimal codes. Moreover, we obtain two new constructions of cutting blocking sets of small cardinality in finite projective spaces. In turn, these allow us to give explicit constructions of minimal codes having short length for the given field and dimension.
