Non-overlapping codes are block codes that have arisen in diverse contexts of computer science and biology. Applications typically require finding non-overlapping codes with large cardinalities, but the maximum size of non-overlapping codes has been determined only for cases where the codeword length divides the size of the alphabet, and for codes with codewords of length two or three. For all other alphabet sizes and codeword lengths no computationally feasible way to identify non-overlapping codes that attain the maximum size has been found to date. Herein we characterize maximal non-overlapping codes. We formulate the maximum non-overlapping code problem as an integer optimization problem and determine necessary conditions for optimality of a non-overlapping code. Moreover, we solve several instances of the optimization problem to show that the hitherto known constructions do not generate the optimal codes for many alphabet sizes and codeword lengths. We also evaluate the number of distinct maximum non-overlapping codes.

非重叠代码是在计算机科学和生物学的不同背景下出现的块代码。应用通常需要找到具有大基数的非重叠代码，但仅在代码字长度除以字母表大小的情况以及代码字长度为二或三的代码的情况下才确定非重叠代码的最大大小。对于所有其他字母大小和码字长度，迄今为止还没有找到计算上可行的方法来识别达到最大大小的非重叠代码。在这里，我们描述了最大非重叠码。我们将最大非重叠码问题表述为整数优化问题，并确定非重叠码最优性的必要条件。此外，我们解决了优化问题的几个实例，以表明迄今为止已知的结构不能为许多字母大小和码字长度生成最佳代码。我们还评估不同的最大非重叠代码的数量。