For a lattice/linear code, we define the Voronoi spherical cumulative density function (CDF) as the CDF of the $\ell_2$-norm/Hamming weight of a random vector uniformly distributed over the Voronoi cell. Using the first moment method together with a simple application of Jensen's inequality, we develop lower bounds on the expected Voronoi spherical CDF of a random lattice/linear code. Our bounds are valid for any finite dimension and are quite close to a trivial ball-based lower bound. They immediately translate to new non-asymptotic upper bounds on the normalized second moment and the error probability of a random lattice over the additive white Gaussian noise channel, as well as new non-asymptotic upper bounds on the Hamming distortion and the error probability of a random linear code over the binary symmetric channel. In particular, we show that for most lattices in $\mathbb{R}^n$ the second moment is greater than that of a Euclidean ball with the same covolume only by a $\left(1+O(\frac{1}{n})\right)$ multiplicative factor. Similarly, for most linear codes in $\mathbb{F}_2^n$ the expected Hamming distortion is greater than that of a corresponding Hamming ball only by an additive universal constant.

翻译：对于格点/线性码，我们定义Voronoi球面累积分布函数（CDF）为均匀分布于Voronoi胞腔的随机向量的$\ell_2$范数/汉明重量的累积分布函数。通过结合一阶矩方法与詹森不等式的简单应用，我们建立了随机格点/线性码的期望Voronoi球面CDF的下界。所得下界适用于任意有限维情形，且非常接近基于平凡球体的下界。这些结果可直接转化为加性高斯白噪声信道下随机格点的归一化二阶矩与误码概率的非渐近上界，以及二元对称信道下随机线性码的汉明失真与误码概率的非渐近上界。特别地，我们证明对于$\mathbb{R}^n$中的大多数格点，其二阶矩仅比具有相同余体积的欧几里得球体的二阶矩大$\left(1+O(\frac{1}{n})\right)$倍乘因子。类似地，对于$\mathbb{F}_2^n$中的大多数线性码，其期望汉明失真仅比对应汉明球体的汉明失真大一个通用的加法常数。