We derive refined entropy upper bounds for $q$-ary $B_2$ codes by exploiting the Fourier structure of the i.i.d. difference distribution $D=X-Y$. Since the pmf of $D$ is an autocorrelation, its Fourier series is a nonnegative trigonometric polynomial of degree at most $q-1$. This leads to a natural convex relaxation over candidate difference distributions, equivalently expressible through an infinite family of positive semidefinite Toeplitz constraints. The resulting formulation admits a simple Gram interpretation and yields certified upper bounds through truncated semidefinite programs. Combined with the prefix-suffix method, this gives improved asymptotic rate upper bounds for $q$-ary $B_2$ codes; in particular, for $q\in\{9,10,11,12,13\}$ the resulting values improve on the best bounds known in the literature. We also study binary constant-weight $B_2$ codes. Extending the distance-distribution method of Cohen, Litsyn, and Zémor to the constant-weight setting, and combining it with Litsyn's asymptotic linear-programming bound for constant-weight codes, we derive a new upper bound on the constant-weight $B_2$ rate.

翻译：我们利用独立同分布差分 $D=X-Y$ 的傅里叶结构，推导了 $q$ 元 $B_2$ 码的更精细熵上界。由于 $D$ 的概率质量函数是自相关函数，其傅里叶级数为次数不超过 $q-1$ 的非负三角多项式。这引出了关于候选差分分布的自然凸松弛，等价地可通过无穷族半正定托普利茨约束来表达。所得公式具有简洁的格拉姆矩阵解释，并通过截断半定规划给出经认证的上界。结合前缀-后缀方法，这改进了 $q$ 元 $B_2$ 码的渐进速率上界；特别地，对于 $q\in\{9,10,11,12,13\}$，所得结果优于文献中已知的最佳界。我们还研究了二元常重 $B_2$ 码。将 Cohen、Litsyn 和 Zémor 的距离分布方法推广至常重情形，并联合 Litsyn 的常重码渐进线性规划界，导出了常重 $B_2$ 码速率的新上界。