Bounds on the Minimum Homogeneous Distance of the ${p^{r}}$-ary Image of Linear Block Codes Over the Galois Ring ${\hbox{GR}}(p^{r},m)$

Solé, Patrick; Sison, Virgilio P.

doi:10.1109/tit.2007.896891

Cited by 12 publications

(19 citation statements)

References 15 publications

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“…This includes the structure of Galois rings as finite chain rings and Frobenius rings and the homogeneous weight that can be defined on them, the construction of Galois ring codes and their images with respect to a modular basis, and the notions of duality and normality of bases of Galois rings which are used in the derivation of optimal image codes. For a more thorough discussion of these topics the reader is referred to [1], [2], [3], [4], and [5].…”

Section: Preliminaries and Definitionsmentioning

confidence: 99%

“…The following proposition in [1] shows how the minimum Hamming weight of B provides a bound for the minimum homogeneous distance of τ (B).…”

Section: Galois Rings and Homogeneous Weightmentioning

confidence: 99%

“…It was determined in [1] that the quaternary image of the 3-quasi cyclic rate-2/6 linear block code over GR(4, 2) with generator matrix 3 + 2ω 2 + 3ω ω ω 3 + 2ω 2 + 3ω ω 3 + 2ω 2 + 3ω 3 + 2ω 2 + 3ω ω…”

Section: Galois Rings and Homogeneous Weightmentioning

confidence: 99%

“…The p r -ary image is a linear block code of length mn over the residue class ring Z p r , and its distance is with respect to the homogeneous metric defined on GR(p r , m) seen as a finite Frobenius chain ring. The main reference for this result is a previous paper [1] which considers for the most part the so-called polynomial basis of GR(p r , m). It is worthy to investigate the dual and normal bases to see how a change in basis affects the parameters of the p r -ary image.…”

Section: Introductionmentioning

confidence: 99%

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Galois ring codes and their images under various bases

Sison

Santos

2018

J. Phys.: Conf. Ser.

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View the article online for updates and enhancements. Abstract. In this paper we consider linear block codes B of length n over the Galois ring GR(p r , m) and obtain their images with respect to various bases of GR(p r , m) seen as a free module of rank m over the residue class ring Zpr . Interesting new examples of dual, normal and self-dual bases of GR(p r , m) and their relationships are given. The image of B is a linear block code over Zpr of length mn and its generator matrix is formed row-wise by the images of βiG, where {βi} m i=1 is a chosen basis of GR(p r , m) and G is a generator matrix of B. Certain conditions in which the p r -ary image is distance-invariant after a change in basis are investigated. Consequently a new quaternary code endowed with a homogeneous metric that is optimal with respect to certain known bounds is constructed. IntroductionThe motivation for this paper came from finding the p r -ary image of a linear block code B of length n over the Galois ring R = GR(p r , m) with respect to any m-basis of R, and placing a bound on its minimum homogeneous distance in terms of the Hamming distance and other code parameters of B. The p r -ary image is a linear block code of length mn over the residue class ring Z p r , and its distance is with respect to the homogeneous metric defined on GR(p r , m) seen as a finite Frobenius chain ring. The main reference for this result is a previous paper [1] which considers for the most part the so-called polynomial basis of GR(p r , m). It is worthy to investigate the dual and normal bases to see how a change in basis affects the parameters of the p r -ary image. For a thorough discussion of the bases of GR(p r , m) over Z p r the reader is referred to [2]. This paper constructed the dual basis using matrix algebra involving the generalized Frobenius automorphism. It showed that a basis of GR(p r , m) is self-dual if and only its automorphism matrix is orthogonal, and is normal if and only its automorphism matrix is symmetric.The material is organized as follows: Section 2 gives the preliminaries and basic definitions while Section 3 gives the main results. Several new illustrative examples are also provided.

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Section: Preliminaries and Definitionsmentioning

confidence: 99%

“…The following proposition in [1] shows how the minimum Hamming weight of B provides a bound for the minimum homogeneous distance of τ (B).…”

Section: Galois Rings and Homogeneous Weightmentioning

confidence: 99%

Section: Galois Rings and Homogeneous Weightmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations