论文标题
双面可靠测试的代码
Two-sided Robustly Testable Codes
论文作者
论文摘要
我们表明，两个随机线性代码的张量产物具有很高的概率可鲁棒测试。这意味着可以获得成对的线性代码，以便它们的产物和双重代码的乘积同时可靠地测试。在渐近良好的量子LDPC代码的最新构建体中，这种两面可靠测试的代码（鲁棒性的弱形式较弱）是确保其线性最小距离的关键成分。我们希望在此处显示的具有更强形式的鲁棒性形式的代码可以用来简化证明并在这些结构中提供更好的距离界限。我们还提供了新的非常简单的不可测试代码的示例。我们表明，如果两个代码的奇偶校验检查是相互正交的，那么它们的产物就无法进行可靠的测试。特别是，这意味着代码具有双重的乘积永远无法妥善测试。我们还研究了称为产品扩展的线性代码集合的属性，该代码可被视为与这些代码产品自然相关的Cochain复合物的串联扩展。我们表明，该属性与代码产品产品的可靠可检验性和一致性可检验性有关。
We show that the tensor product of two random linear codes is robustly testable with high probability. This implies that one can obtain pairs of linear codes such that their product and the product of their dual codes are simultaneously robustly testable. Such two-sided robustly testable codes (with a much weaker form of robustness) were the key ingredient in the recent constructions of asymptotically good quantum LDPC codes, which ensured their linear minimum distance. We hope that the existence of such codes with a stronger form of robustness, shown here, can be used to simplify the proofs and provide better distance bounds in these constructions. We also give new very simple examples of non-robustly testable codes. We show that if the parity-checks of two codes are mutually orthogonal, then their product is not robustly testable. In particular, this implies that the product of a code with its dual can never be robustly testable. We also study a property of a collection of linear codes called product-expansion, which can be viewed as a coboundary expansion of the cochain complex naturally associated with the product of these codes. We show that this property is related with the robust testability and the agreement testability of the products of codes.
