On codes over Frobenius rings: generating characters, MacWilliams identities and generator matrices

Dougherty, Steven T.; Salturk, Esengül; Szabo, Steve

doi:10.1007/s00200-019-00384-0

Cited by 6 publications

(2 citation statements)

References 10 publications

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“…Martinez-Moro and Szabo [4] gave the 21 commutative local rings with identity of order 16. Of these, 15 are Frobenius, where 8 are chain rings and 7 are non-chain rings, as given in [5]. The ring R belongs to the class of finite 16-element Frobenius commutative rings with identity which are local, non-chain, and non-principal ideal rings.…”

Section: B Linear Block Codesmentioning

confidence: 99%

Isometries and binary images of linear block codes over ℤ₄ + uℤ₄ and ℤ₈ + uℤ₈

Sison

Remillion

2017

J. Phys.: Conf. Ser.

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Let F2 be the binary field and Z2r the residue class ring of integers modulo 2 r , where r is a positive integer. For the finite 16-element commutative local Frobenius nonchain ring Z4 + uZ4, where u is nilpotent of index 2, two weight functions are considered, namely the Lee weight and the homogeneous weight. With the appropriate application of these weights, isometric maps from Z4 + uZ4 to the binary spaces F 4 2 and F 8 2 , respectively, are established via the composition of other weight-based isometries. The classical Hamming weight is used on the binary space. The resulting isometries are then applied to linear block codes over Z4 + uZ4 whose images are binary codes of predicted length, which may or may not be linear. Certain lower and upper bounds on the minimum distances of the binary images are also derived in terms of the parameters of the Z4 + uZ4 codes. Several new codes and their images are constructed as illustrative examples. An analogous procedure is performed successfully on the ring Z8 + uZ8, where u 2 = 0,which is a commutative local Frobenius non-chain ring of order 64. It turns out that the method is possible in general for the class of rings Z2r + uZ2r , where u 2 = 0, for any positive integer r, using the generalized Gray map from Z2r to F 2 r−1 2 . Index Terms-Frobenius ring, homogeneous weight, linear block code, binary image.

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Section: B Linear Block Codesmentioning

confidence: 99%

Isometries and binary images of linear block codes over ℤ₄ + uℤ₄ and ℤ₈ + uℤ₈

Sison

Remillion

2017

J. Phys.: Conf. Ser.

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“…We proceed to investigate, in this article, two crucial tools in coding theory: MacWilliams relations and generator matrices. In [17], the authors discussed these tools over local Frobenius rings with small order, i.e., 16 Frobenius rings with order 16 provided in [18]. Thus, we aim to concentrate on rings of order 32 and utilize them as examples.…”

Section: Introductionmentioning

confidence: 99%

On Linear Codes over Finite Singleton Local Rings

Alabiad,

Alhomaidhi,

Alsarori

2024

Mathematics

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The study of linear codes over local rings, particularly non-chain rings, imposes difficulties that differ from those encountered in codes over chain rings, and this stems from the fact that local non-chain rings are not principal ideal rings. In this paper, we present and successfully establish a new approach for linear codes of any finite length over local rings that are not necessarily chains. The main focus of this study is to produce generating characters, MacWilliams identities and generator matrices for codes over singleton local Frobenius rings of order 32. To do so, we first start by characterizing all singleton local rings of order 32 up to isomorphism. These rings happen to have strong connections to linear binary codes and Z4 codes, which play a significant role in coding theory.

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MacWilliams Identities and Generator Matrices for Linear Codes over ℤp4[u]/(u2 − p3β, pu)

Alabiad,

Alhomaidhi,

Alsarori

2024

Axioms

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Suppose that R=Zp4[u] with u2=p3β and pu=0, where p is a prime and β is a unit in R. Then, R is a local non-chain ring of order p5 with a unique maximal ideal J=(p,u) and a residue field of order p. A linear code C of length N over R is an R-submodule of RN. The purpose of this article is to examine MacWilliams identities and generator matrices for linear codes of length N over R. We first prove that when p≠2, there are precisely two distinct rings with these properties up to isomorphism. However, for p=2, only a single such ring is found. Furthermore, we fully describe the lattice of ideals of R and their orders. We then calculate the generator matrices and MacWilliams relations for the linear codes C over R, illustrated with numerical examples. It is important to address that there are challenges associated with working with linear codes over non-chain rings, as such rings are not principal ideal rings.

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On codes over Frobenius rings: generating characters, MacWilliams identities and generator matrices

Cited by 6 publications

References 10 publications

Isometries and binary images of linear block codes over ℤ₄ + uℤ₄ and ℤ₈ + uℤ₈

Isometries and binary images of linear block codes over ℤ₄ + uℤ₄ and ℤ₈ + uℤ₈

On Linear Codes over Finite Singleton Local Rings

MacWilliams Identities and Generator Matrices for Linear Codes over ℤp4[u]/(u2 − p3β, pu)

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On codes over Frobenius rings: generating characters, MacWilliams identities and generator matrices

Cited by 6 publications

References 10 publications

Isometries and binary images of linear block codes over ℤ4 + uℤ4 and ℤ8 + uℤ8

Isometries and binary images of linear block codes over ℤ4 + uℤ4 and ℤ8 + uℤ8

On Linear Codes over Finite Singleton Local Rings

MacWilliams Identities and Generator Matrices for Linear Codes over ℤp4[u]/(u2 − p3β, pu)

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Product

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Isometries and binary images of linear block codes over ℤ₄ + uℤ₄ and ℤ₈ + uℤ₈

Isometries and binary images of linear block codes over ℤ₄ + uℤ₄ and ℤ₈ + uℤ₈