A fundamental problem in quantum coding theory is to determine the maximum size of quantum codes of given block length and distance. A recent work introduced bounds based on semidefinite programming, strengthening the well-known quantum linear programming bounds. However, floating-point inaccuracies prevent the extraction of rigorous non-existence proofs from the numerical methods. Here, we address this by providing rational infeasibility certificates for a range of quantum codes. Using a clustered low-rank solver with heuristic rounding to algebraic expressions, we can improve upon $18$ upper bounds on the maximum size of $n$-qubit codes with $6 \leq n \leq 19$. Our work highlights the practicality and scalability of semidefinite programming for quantum coding bounds.

翻译：量子编码理论中的一个基本问题是确定给定码长与距离的量子码的最大尺寸。近期的一项研究引入了基于半定规划（SDP）的界，强化了著名的量子线性规划界。然而，浮点数不精确性使得无法从数值方法中提取严格的非存在性证明。本文通过为一系列量子码提供有理数不可行性证明来解决这一问题。采用聚类低秩求解器结合启发式有理代数表达式舍入方法，我们改进了$6 \leq n \leq 19$范围内$n$量子比特码最大尺寸的$18$个上界。本文突显了半定规划在量子码界问题中的实用性与可扩展性。