The q-ary image of a q/sup m/-ary cyclic code

Seguin, G.

doi:10.1109/18.370140

Cited by 22 publications

(14 citation statements)

References 10 publications

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“…can be easily obtained. Note that we do not attempt to address cyclic permutations of the binary images, as was done previously by Seguin [3].…”

Section: Introductionmentioning

confidence: 98%

“…[2], specifically for RS binary images. The decomposition structure of RS binary images has been formulated in different ways [2], [3], [4], [5], which are essentially alike. We first review the material in [2] before stating our new results.…”

Section: Introductionmentioning

confidence: 99%

“…The matrices in (1) and (3) are m × m matrices, where the ith column of the matrix corresponds to a single basis element γ i . In relation to the matrix structures, we define the tuple (i, j) that indexes the ith column and the jth bit within the column.…”

Section: Introductionmentioning

confidence: 99%

“…Also, the binary-images of RS codes are of particular interest, for transmission over practical communication systems. The structure of the binary-image of a cyclic code has been widely studied [1], [2], [3], [4], and it has been shown that it is possible to derive automorphism groups from these structures [2], [3].…”

Section: Introductionmentioning

confidence: 99%

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Notes on the automorphism groups of Reed Solomon binary images

Lim

Fossorier

Kavcic

2008

2008 IEEE International Symposium on Information Theory

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In this paper, a new proof of the work by Lacan et. al is obtained. This approach led to newly discovered connections between the automorphism group of the Reed Solomon (RS) binary image, and elementary group theory. This new development facilitated proving the existence and constructing permutations that have not been previously reported in the literature. Simple special cases are then considered in this work.

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“…can be easily obtained. Note that we do not attempt to address cyclic permutations of the binary images, as was done previously by Seguin [3].…”

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Notes on the automorphism groups of Reed Solomon binary images

Lim

Fossorier

Kavcic

2008

2008 IEEE International Symposium on Information Theory

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“…A few problems have attracted some attention in the past. The problem of determining when the q-ary image of a cyclic code over GF(q m ) is cyclic was solved in [7] by using a module structure for images. Perhaps the most interesting problem related to images is the determination of minimum distance of the image of a code.…”

mentioning

confidence: 99%

Self-Orthogonality of $q$-Ary Images of $q^{m}$-Ary Codes and Quantum Code Construction

Sundeep

Thangaraj

2007

IEEE Trans. Inform. Theory

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A code over GF(q m ) can be imaged or expanded into a code over GF(q) using a basis for the extension field over the base field. The properties of such an image depend on the original code and the basis chosen for imaging.Problems relating the properties of a code and its image with respect to a basis have been of great interest in the field of coding theory. In this work, a generalized version of the problem of self-orthogonality of the q-ary image of a q m -ary code has been considered. Given an inner product (more generally, a biadditive form), necessary and sufficient conditions have been derived for a code over a field extension and an expansion basis so that an image of that code is self-orthogonal. The conditions require that the original code be self-orthogonal with respect to several related biadditive forms whenever certain power sums of the dual basis elements do not vanish. Numerous interesting corollaries have been derived by specializing the general conditions. An interesting result for the canonical or regular inner product in fields of characteristic two is that only self-orthogonal codes result in self-orthogonal images. Another result is that image of a code is self-orthogonal for all bases if and only if trace of the code is self-orthogonal, except for the case of binary images of 4-ary codes. The conditions are particularly simple to state and apply for cyclic codes. To illustrate a possible application, new quantum error-correcting codes have been constructed with larger minimum distance than previously known.

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On the q-ary image of cyclic codes

Jensen

1997

Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

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The q-ary image of a q/sup m/-ary cyclic code

Cited by 22 publications

References 10 publications

Notes on the automorphism groups of Reed Solomon binary images

Notes on the automorphism groups of Reed Solomon binary images

Self-Orthogonality of $q$-Ary Images of $q^{m}$-Ary Codes and Quantum Code Construction

On the q-ary image of cyclic codes

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