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Monomial isomorphisms of cyclic codes

机译：循环码的单项同构

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摘要

For codes, there are multiple notions of isomorphism. For example, we can consider isomorphisms that only permute the coordinates of codewords, or isomorphisms that not only permute the coordinates of codewords but also multiply each coordinate by a scalar (not necessarily the same scalar for each coordinate) as it permutes the coordinates. Isomorphisms of cyclic codes of the first kind have been studied in some circumstances-we will call them permutation isomorphisms-and our purpose is to begin the study of the second kind of isomorphism-which we call monomial isomorphisms-for cyclic codes. We give examples of cyclic codes that are monomially isomorphic but not permutationally isomorphic. We also show that the monomial isomorphism problem for cyclic codes of length over reduces to the permutation isomorphism problem for cyclic codes of length over if and only if . Applying known results, this solves the monomial isomorphism problem for cyclic codes satisfying . Additionally, we solve the monomial isomorphism problem for cyclic codes of prime length over all finite fields. Finally, our results also hold for some codes that are not cyclic.
机译：对于代码，有多个同构概念。例如，我们可以考虑仅置换代码字坐标的同构，或考虑不仅置换代码字坐标而且将每个坐标乘以标量（对于每个坐标不一定是相同的标量）的同构。在某些情况下已经研究了第一类循环码的同构，我们将其称为置换同构，而我们的目的是开始研究第二种同构，即循环码的单项同构。我们给出了单调同构但不是置换同构的循环码示例。我们还表明，长度为大于的循环码的单项同构问题可以简化为长度为大于的循环码的置换同构问题。应用已知结果，这解决了满足的循环码的单项同构问题。另外，我们解决了所有有限域上素数长度的循环码的单项同构问题。最后，我们的结果还适用于某些非循环代码。

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