In this paper, we study binary constrained codes that are resilient to bit-flip errors and erasures. In our first approach, we compute the sizes of constrained subcodes of linear codes. Since there exist well-known linear codes that achieve vanishing probabilities of error over the binary symmetric channel (which causes bit-flip errors) and the binary erasure channel, constrained subcodes of such linear codes are also resilient to random bit-flip errors and erasures. We employ a simple identity from the Fourier analysis of Boolean functions, which transforms the problem of counting constrained codewords of linear codes to a question about the structure of the dual code. We illustrate the utility of our method in providing explicit values or efficient algorithms for our counting problem, by showing that the Fourier transform of the indicator function of the constraint is computable, for different constraints. Our second approach is to obtain good upper bounds, using an extension of Delsarte's linear program (LP), on the largest sizes of constrained codes that can correct a fixed number of combinatorial errors or erasures. We observe that the numerical values of our LP-based upper bounds beat the generalized sphere packing bounds of Fazeli, Vardy, and Yaakobi (2015).

翻译：本文研究能够抵抗比特翻转错误与擦除错误的二进制约束码。第一种方法中，我们计算线性码的约束子码的大小。由于存在已知的线性码能在二进制对称信道（导致比特翻转错误）和二进制擦除信道上实现错误概率趋近于零，这些线性码的约束子码同样能抵抗随机比特翻转错误与擦除。我们利用布尔函数傅里叶分析中的一个简单恒等式，将约束码字计数问题转化为对偶码结构问题。通过证明不同约束条件下约束指示函数的傅里叶变换是可计算的，我们展示了该方法在提供显式数值或高效计数算法方面的实用性。第二种方法基于推广德尔萨特线性规划，对能纠正固定数量组合错误或擦除的约束码的最大尺寸给出了良好上界。我们发现基于线性规划上界的数值结果优于法泽利、瓦尔迪和亚科比（2015）提出的广义球堆积界。