AI 中文总结
研究如何利用汉明码构造覆盖序列及覆盖序列码，针对小半径情况构建短长度序列和相关码，还构造了长度渐近最优的小半径序列及较大半径的有趣码。
AI 中文摘要
一个\((n,R)\)覆盖序列是一个循环序列，其连续的\(n\)元组构成一个长度为\(n\)、覆盖半径为\(R\)的码。一个\((n,m,R)\)覆盖序列码是一组长度为\(m\)的循环序列，其连续的\(n\)元组构成一个长度为\(n\)、覆盖半径为\(R\)的码。这些码是\((n,R)\)覆盖序列的最佳构建块。我们展示了对于小半径情况，如何使用汉明码来构造短长度的此类序列以及具有相对较少序列数和码字总数的此类码。构造了长度渐近接近最优的小半径序列，特别是对于足够大的素数幂大小的字母表。通过相同构造，也为较大半径构造了有趣的码。
英文摘要
An $(n,R)_q$-covering sequence is a cyclic sequence, over the finite field $\F_q$, whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. An $(n,m,R)_q$-covering-sequences code is a set of cyclic sequences of length $m$, over $\F_q$, whose consecutive $n$-tuples form a code of length $n$ and covering radius $R$. These codes are the best building blocks for $(n,R)_q$-covering sequences. We show, for small radii, how cyclic codes and constacyclic codes with small covering radius, can be used to construct such sequences of short length and such codes with a relatively small number of sequences and a total number of codewords in the associated covering code. Sequences with small radius whose length approaches asymptotically to optimality are constructed, especially for an alphabet of prime power size large enough. With the same construction, interesting codes are also constructed for larger radii.
