Abstract
Let R be a (possibly noncommutative) finite principal ideal ring. Via a total ordering of the ring elements and an ordered basis a lexicographic ordering of the module \(R^n\) is produced. This is used to set up a greedy algorithm that selects vectors for which all linear combinations with the previously selected vectors satisfy a pre-specified selection property and updates the to-be-constructed code to the linear hull of the vectors selected so far. The output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. In this paper we investigate the properties of such lexicodes over finite principal ideal rings and show that the total ordering of the ring elements has to respect containment of ideals for the algorithm to produce meaningful results. Only then it is guaranteed that the algorithm is exhaustive and thus produces codes that are maximal with respect to inclusion. It is further illustrated that the output of the algorithm heavily depends on the total ordering and chosen basis.
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Notes
Note that the definition of respectfulness for an ordering is based on principal left ideals. If we simply ignore the non-principal ideal (x, y) and follow Definition 3.1, then any ordering of the form \(0<1<\text {``rest''}\) may be called respectful.
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Acknowledgements
We wish to thank the reviewers for their close reading. In particular the suggestion of how to present selection properties and to rewrite the proofs of Theorems 3.2 and 4.4 has led to a more concise version of the paper. H. Gluesing-Luerssen was partially supported by the National Science Foundation Grant DMS-1210061 and by the Grant #422479 from the Simons Foundation.
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Communicated by J.-L. Kim.
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Antrobus, J., Gluesing-Luerssen, H. Lexicodes over finite principal ideal rings. Des. Codes Cryptogr. 86, 2661–2676 (2018). https://doi.org/10.1007/s10623-018-0469-2
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DOI: https://doi.org/10.1007/s10623-018-0469-2