Codes and Projective Multisets

Dodunekov, S.M.; Simonis, Juriaan

doi:10.37236/1375

Cited by 81 publications

(103 citation statements)

References 29 publications

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“…We say that the multiset k S and the code C = c∈S Rc are associated. By definition of k S we have |k S | = n. The following theorem is a generalization of a similar result by Dodunekov and Simonis [10] about linear codes over finite fields. Proof.…”

Section: Multisets In Projective Hjelmslev Geometries and Linear Codementioning

confidence: 53%

“…Codes over such rings appeared in various contexts in recent coding theory research. In third place, nontrivial linear codes over finite chain rings can be considered as multisets of points in finite projective Hjelmslev geometries thus extending the familiar interpretation of linear codes over finite fields as multisets of points in classical projective geometries PG(k, q) [10]. However, there are some differences between linear codes over finite fields and linear codes over finite chain rings.…”

Section: Introductionmentioning

confidence: 99%

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Linear Codes over Finite Chain Rings

Honold¹,

Landjev²

1999

Electron. J. Combin.

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The aim of this paper is to develop a theory of linear codes over finite chain rings from a geometric viewpoint. Generalizing a well-known result for linear codes over fields, we prove that there exists a one-to-one correspondence between so-called fat linear codes over chain rings and multisets of points in projective Hjelmslev geometries, in the sense that semilinearly isomorphic codes correspond to equivalent multisets and vice versa. Using a selected class of multisets we show that certain MacDonald codes are linearly representable over nontrivial chain rings.

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Section: Multisets In Projective Hjelmslev Geometries and Linear Codementioning

confidence: 53%

Section: Introductionmentioning

confidence: 99%

Linear Codes over Finite Chain Rings

Honold¹,

Landjev²

1999

Electron. J. Combin.

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“…For the corresponding Frobenius number the sharpest upper bound in the binary case q = 2 isF 2 (r) ≤ 2 2r − 2 r−1 − 1. The lengths of projective 2-, 4-divisible and 8-divisible linear binary codes as well as 3-divisible linear ternary codes have been completely determined [14,20], but there are open cases already for (q, ∆) = (2, 16), (3,9), (5,5).…”

Section: Resultsmentioning

confidence: 99%

On the Lengths of Divisible Codes

Kiermaier

Kurz

2020

IEEE Trans. Inform. Theory

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In this article, the effective lengths of all q r -divisible linear codes over F q with a non-negative integer r are determined. For that purpose, the S q (r)-adic expansion of an integer n is introduced. It is shown that there exists a q r -divisible F q -linear code of effective length n if and only if the leading coefficient of the S q (r)-adic expansion of n is non-negative. Furthermore, the maximum weight of a q r -divisible code of effective length n is at most σ q r , where σ denotes the cross-sum of the S q (r)-adic expansion of n.This result has applications in Galois geometries. A recent theorem of Nȃstase and Sissokho on the maximum sizes of partial spreads follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.1 By [31, Th. 1], for ∆ = p e d with p the characteristic of the base field F q and p d, each full-length ∆divisible F q -linear code is the d-fold repetition of a p e -divisible F q -linear code.

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“…Obviously, the x v are non-negative integers. The conditions (1) and (5) correspond to the restriction that the weights are ∆-divisible and contained in {a∆, . .…”

Section: Appendix a An Alternative Computational Approachmentioning

confidence: 99%