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On homogeneous arcs and linear codes over finite chain rings

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Abstract

We consider projective Hjelmslev geometries over finite chain rings of length 2 with residue field of order q. In these geometries we introduce and investigate the so-called homogeneous arcs defined as multisets of points with respect to a special weight function. These arcs are associated with linearly representable q-ary codes. We establish a relation between the parameters of a linearly representable code and the parameters of the associated arc. We prove an inequality for the largest homogeneous weight of an arc of given size N. We consider the \(\tau \)-dual of a homogeneous arc and give an explicit formula for the homogeneous weights of the hyperplanes with respect to the dual arc. We prove that if a homogeneous arc is of constant weight, this weight is forced to be zero and the arc is a sum of neighbour classes of points. We prove various numerical conditions on the parameters of homogeneous two-weight arcs, as well as an interesting identity involving the parameters of a projective homogeneous two-weight arc and its dual, which isturns out to be also a homogeneous two-weight arc. Finally, we present examples of two-weight homogeneous arcs some of which are new.

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Notes

  1. In [10, 11] the term “fat” was used in place of “regular”; the latter has been adopted in [3] and used since then.

  2. Except for the trivial case of a torsion module M, these codes are not projective either.

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Acknowledgements

The work of the first author was supported by the National Natural Science Foundation of China under Grant 61571006. The work of the second author was supported by the Bulgarian National Science Research Fund under Contract K\(\varPi \)-06-32/6—07.12.2019. The authors thank the annonimous referees for their careful reading and the valuable comments that helped to improve this paper.

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Authors and Affiliations

  1. ZJU-UIUC Institute, International Campus of Zhejiang University, 718 East Haizhou Road, Haining, 314400, China

    Thomas Honold

  2. Department of Information and Electronic Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, 310027, China

    Thomas Honold

  3. New Bulgarian University, 21 Montevideo Str., 1618, Sofia, Bulgaria

    Ivan Landjev

  4. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 Acad G. Bonchev Str., 1113, Sofia, Bulgaria

    Ivan Landjev

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  1. Thomas Honold
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  2. Ivan Landjev
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Correspondence to Ivan Landjev.

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Honold, T., Landjev, I. On homogeneous arcs and linear codes over finite chain rings. AAECC 34, 359–375 (2023). https://doi.org/10.1007/s00200-021-00501-y

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  • DOI: https://doi.org/10.1007/s00200-021-00501-y

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