Repeated root Cyclic and Negacyclic codes over Galois rings have been studiedmuch less than their simple root counterparts. This situation is beginning tochange. For example, repeated root codes of length $p^s$, where $p$ is thecharacteristic of the alphabet ring, have been studied under some additionalhypotheses. In each one of those cases, the ambient space for the codes hasturned out to be a chain ring. In this paper, all remaining cases of cyclic andnegacyclic codes of length $p^s$ over a Galois ring alphabet are considered. Inthese cases the ambient space is a local ring with simple socle but not a chainring. Nonetheless, by reducing the problem to one dealing with uniserialsubambients, a method for computing the Hamming distance of these codes isprovided.
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机译：已经研究了在Galois环上重复的根循环码和负循环码，而不是简单的根对应物。这种情况开始改变。例如，已经研究了长度为$ p ^ s $的重复根代码，其中$ p $是字母环的特征，在某些其他假设下也进行了研究。在每种情况下，代码的周围空间都变成了链环。在本文中，考虑了Galois环字母表上长度为$ p ^ s $的循环和负循环码的所有剩余情况。在这种情况下，周围空间是具有简单鞋底但不是链环的局部环。尽管如此，通过将问题简化为处理无意义的环境，提供了一种计算这些代码的汉明距离的方法。
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