Generalized quadratic residue codes

Lint, JH van; MacWilliams, F. J.

doi:10.1109/tit.1978.1055965

Cited by 45 publications

(22 citation statements)

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“…Table V gives the first codes obtained by these constructions. The [38, 19,12] and [46,23,14] codes are new and their minimum distance exceeds the highest distance previously known for these parameters. The minimum distance 16 of the codes of lengths 58 and 62 is from [8], and this is the highest minimum distance known for self-dual codes of these lengths.…”

Section: Quadratic Double Circulant Over Gf(5)mentioning

confidence: 96%

“…Of course in the case where q is not a prime the matrix Q q (r, s, t) is not circulant; it corresponds to the case of generalized quadratic residue codes [14], but we keep the same name for simplicity. We also define for any u non null and v in R the [2q+2, q+1] matrix:…”

Section: Definitionsmentioning

confidence: 99%

“…The codes of lengths 34 and higher are new. In particular the [46,23,14], [48,24,14], [62,31,16], and [64, 32,16] codes are new and have a better minimum weight than any previously known codes with these lengths and dimensions (cf tables with best known minimum weight of [4] [7]. Table II lists the codes built for the first lengths, a ?…”

Section: =(4k − 1) I+(j − I)(6k − 2)=i and We Therefore Obtain A Selmentioning

confidence: 99%

“…In particular we give new infinite constructions for self-dual codes over GF(4) (for which nothing was known), GF (5), GF (7) and GF (9); some of them lead to new codes with better parameters than previously known codes. Examples of such codes are [46,23,14], [48,24,14], [62,31,16], and [64,32,16] codes over GF (4). We prove that for some parameters, as in the binary and ternary cases, the automorphism groups of QDC codes contain the group PSL 2 (q).…”

Section: Introductionmentioning

confidence: 99%

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Quadratic Double Circulant Codes over Fields

Gaborit

2002

Journal of Combinatorial Theory, Series A

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Section: Quadratic Double Circulant Over Gf(5)mentioning

confidence: 96%

Section: Definitionsmentioning

confidence: 99%

Section: =(4k − 1) I+(j − I)(6k − 2)=i and We Therefore Obtain A Selmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quadratic Double Circulant Codes over Fields

Gaborit

2002

Journal of Combinatorial Theory, Series A

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“…In particular, lattice No. 10 comes from a quadratic residue code [13,23] of length 14. Now we come to a particular construction of a unimodular lattice of rank 15.…”

Section: Finallymentioning

confidence: 99%

Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers

Abdukhalikov

Scharlau

2009

Math. Comp.

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Abstract. All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in Q( √ −3) are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3.In 1978 W. Feit [10] classified the unimodular hermitian lattices of dimensions up to 12 over the ring Z[ω] of Eisenstein integers, where ω is a primitive third root of unity. These lattices all have roots, that is, vectors of norm 2. In dimension 13, for the first time a unimodular lattice without roots appears [1,3]. In [2] the unimodular lattices in dimension 13 are completely classified. The root-free lattice turns out to be unique. It has minimal norm 3, and its automorphism group is isomorphic to the group Z 6 × PSp 6 (3) of order 2 10 .3 10 .5.7.13. The remaining lattices all have roots; the rank of the root system is 12 in all cases.In this paper, we classify the unimodular lattices in dimensions 14 and 15. There are exactly 58, respectively 259 classes of indecomposable lattices in these dimensions. Below, we list their root systems and the orders of their automorphism groups. Gram matrices for all lattices are available electronically via www.mathematik.uni-dortmund.de/~scharlauThere is only one root-free unimodular lattice of rank 14, and there are two root-free unimodular lattices of rank 15.The lattices without roots have minimal norm 3; they are extremal as introduced for unimodular Eisenstein lattices in [8], Chapter 10.7. They give rise to 3-modular extremal Z-lattices in twice the dimension, as defined by Quebbemann in [17]. See [8,19,20] for more information on extremal and modular lattices and their relation to modular forms. In this context, the lattices classified in this paper can be considered as complex structures on (extremal) 3-modular lattices. The question for existence, uniqueness, and possibly a full classification of extremal modular lattices has been an ongoing challenge, both computationally and theoretically, after the appearance of the influential paper [17].Let V be a vector space over Q( √ −3) with a positive definite hermitian product (, ). A lattice L in V is a finitely generated Z[ω]-module contained in V such that L contains a basis of V and (x, y) ∈ Z[ω] for all x, y ∈ L. More precisely, one

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