In a recent note we derived information theoretic upper bounds on the rate of $q$-ary $2$-separable codes and on the rate of $q$-ary $B_2$ codes based on an entropy argument applied coordinate-wise to a suitable pair of random suffixes of codewords. In the $B_2$ case, the bound was obtained by maximizing the entropy of the difference $X-Y$ of two independent $q$-ary random variables under the sole constraint $\mathbb{P}(X=Y)\geq 1/q$. In this paper we refine this step by exploiting the full Fourier-analytic structure of the difference distribution $X-Y$. More precisely, we use that the pmf of $X-Y$ is an autocorrelation of a probability mass function on $\{0,\dots,q-1\}$ and therefore its Fourier transform is a nonnegative trigonometric polynomial of prescribed degree. This leads to a natural convex optimization problem over the coefficients of such polynomials whose optimal value yields a strictly smaller upper bound on the entropy of $X-Y$ and, in turn, to improved bounds on the rate of $q$-ary $B_2$ codes. We evaluate the resulting bound numerically via truncated Toeplitz SDPs and show that for $q\in\{9,10,11,12,13\}$ the new rate upper bounds improve upon the best available bounds in the literature.

翻译：在近期的一篇笔记中，我们基于信息论方法，通过对码字一对合适随机后缀的逐坐标熵论证，推导了 $q$ 元 $2$ 可分离码的速率上界和 $q$ 元 $B_2$ 码的速率上界。在 $B_2$ 情形下，该上界是通过在唯一约束 $\mathbb{P}(X=Y)\geq 1/q$ 下最大化两个独立 $q$ 元随机变量之差 $X-Y$ 的熵得到的。本文通过充分利用差分布 $X-Y$ 的完整傅里叶分析结构来改进这一步骤。具体而言，我们利用 $X-Y$ 的概率质量函数是 $\{0,\dots,q-1\}$ 上某个概率质量函数的自相关这一特性，因此其傅里叶变换为指定次数的非负三角多项式。这导出了一个关于此类多项式系数的自然凸优化问题，其最优值给出了 $X-Y$ 熵的严格更小上界，进而改进了 $q$ 元 $B_2$ 码的速率上界。我们通过截断托普利茨半定规划对所得上界进行数值评估，结果表明，对于 $q\in\{9,10,11,12,13\}$，新的速率上界优于文献中现有的最优上界。