On the additive (/spl Zopf//sub 4/-linear and non-/spl Zopf//sub 4/-linear) Hadamard codes: rank and kernel

Phelps, Kevin T.; Rifà, Josep; Villanueva, Mercè

doi:10.1109/tit.2005.860464

Cited by 54 publications

(92 citation statements)

References 8 publications

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“…We also know Z 2 Z 4 -linear codes such that the rank is in between these two bounds such as, for example, the Hadamard Z 4 -linear codes ( [23] or Example 1). [17].…”

Section: Rank Of Z 2 Z 4 -Additive Codesmentioning

confidence: 99%

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$${\mathbb{Z}_2\mathbb{Z}_4}$$ -linear codes: rank and kernel

Fernández-Córdoba

Pujol

Villanueva

2009

Des. Codes Cryptogr.

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A code C is Z 2 Z 4 -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of Z 2 Z 4 -additive codes under an extended Gray map are called Z 2 Z 4 -linear codes. In this paper, the invariants for Z 2 Z 4 -linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of Z 2 Z 4 -linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a Z 2 Z 4 -linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a Z 2 Z 4 -linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a Z 2 Z 4 -linear code for each possible pair (r, k) is given.

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“…We also know Z 2 Z 4 -linear codes such that the rank is in between these two bounds such as, for example, the Hadamard Z 4 -linear codes ( [23] or Example 1). [17].…”

Section: Rank Of Z 2 Z 4 -Additive Codesmentioning

confidence: 99%

“…The Hadamard Z 2 Z 4 -linear codes H are the Z 2 Z 4 -dual of the extended 1-perfect Z 2 Z 4 -linear codes. The rank of Hadamard Z 2 Z 4 -linear codes was computed in [23] and the rank of extended 1-perfect Z 2 Z 4 -linear codes in [7]. Specifically,…”

Section: Example 1 For Any Integer T ≥ 3 and Eachmentioning

confidence: 99%

$${\mathbb{Z}_2\mathbb{Z}_4}$$ -linear codes: rank and kernel

Fernández-Córdoba

Pujol

Villanueva

2009

Des. Codes Cryptogr.

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“…For example, Z 2 Z 4 -linear perfect single error-correcting codes (or 1-perfect codes) are found in [22] and fully characterized in [6]. Also, in subsequent papers [5,13,14,19,20], Z 2 Z 4 -linear extended perfect and Hadamard codes are studied and classified independently for˛D 0 and˛6 D 0. Finally, in [17,21,23], Z 2 Z 4 -linear Reed-Muller codes are also studied.…”

Section: Introductionmentioning

confidence: 99%

“…In general, C can be written as the union of cosets of Ker.C /, and Ker.C / is the largest linear code for which this is true [2]. The Z 2 Z 4 -linear Hadamard codes can also be classified using either the rank or the dimension of the kernel, as it is proven in [14,20], where these parameters are computed.…”

Section: Introductionmentioning

confidence: 99%

On the Automorphism Groups of the $$\mathbb{Z}_{2}\mathbb{Z}_{4}$$ -Linear Hadamard Codes and Their Classification

Krotov

Villanueva

2015

Coding Theory and Applications

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“…For example, Z 2 Z 4 -linear perfect single error-correcting codes (or 1-perfect codes) are found in [20] and fully characterized in [8]. Also, in subsequent papers [7,15,19,23], Z 2 Z 4 -linear extended 1-perfect and Hadamard codes are studied and classified.…”

mentioning

confidence: 99%

$${{{\mathbb Z}_2}{{\mathbb Z}_4}}$$ -linear codes: generator matrices and duality

Borges

Fernández-Córdoba

Pujol

et al. 2009

Des. Codes Cryptogr.

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148

141

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A code C is Z 2 Z 4 -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). In this paper Z 2 Z 4 -additive codes are studied. Their corresponding binary images, via the Gray map, are Z 2 Z 4 -linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity-check matrices are given. In order to do this, the appropriate concept of duality for Z 2 Z 4 -additive codes is defined and the parameters of their dual codes are computed.

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On the additive (/spl Zopf//sub 4/-linear and non-/spl Zopf//sub 4/-linear) Hadamard codes: rank and kernel

Cited by 54 publications

References 8 publications

$${\mathbb{Z}_2\mathbb{Z}_4}$$ -linear codes: rank and kernel

$${\mathbb{Z}_2\mathbb{Z}_4}$$ -linear codes: rank and kernel

On the Automorphism Groups of the $$\mathbb{Z}_{2}\mathbb{Z}_{4}$$ -Linear Hadamard Codes and Their Classification

$${{{\mathbb Z}_2}{{\mathbb Z}_4}}$$ -linear codes: generator matrices and duality

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