论文标题
随机线性代码和Krawchouk多项式的重量分布
Weight distribution of random linear codes and Krawchouk polynomials
论文作者
论文摘要
对于$ 0 <λ<1 $和$ n \ rightarrow \ infty $以$ \ {0,1 \}^n $的随机$λn$ vectors均匀地挑选，让$ c $是其跨度的正交补充。给定$ 0 <γ<\ frac12 $，带有$ 0 <λ<h（γ）$，让$ x $是随机变量，它计数hamming权重$ i =γn$的单词数（其中$ i $假定为偶数整数）。 Linial和Mosheiff确定了所有订单的$ x $的矩差异（\ frac {n} {\ log n} \ right）$。在本文中，我们将其估计值扩展到线性顺序的时刻。我们的主要观察结果是，适当归一化的$ k^{th} $时刻$ x $的行为实际上取决于krawchouk polyenmial $ k_i $的$ k^{th} $ norm。
For $0 < λ< 1$ and $n \rightarrow \infty$ pick uniformly at random $λn$ vectors in $\{0,1\}^n$ and let $C$ be the orthogonal complement of their span. Given $0 < γ< \frac12$ with $0 < λ< h(γ)$, let $X$ be the random variable that counts the number of words in $C$ of Hamming weight $i = γn$ (where $i$ is assumed to be an even integer). Linial and Mosheiff determined the asymptotics of the moments of $X$ of all orders $o\left(\frac{n}{\log n}\right)$. In this paper we extend their estimates up to moments of linear order. Our key observation is that the behavior of the suitably normalized $k^{th}$ moment of $X$ is essentially determined by the $k^{th}$ norm of the Krawchouk polynomial $K_i$.
