Algebraic techniques for nonlinear codes

Bauer, Heinz; Ganter, Bernhard; Hergert, F.

doi:10.1007/bf02579339

Cited by 61 publications

(41 citation statements)

References 2 publications

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“…We have already ruled out 2 k -k -2 by our previous argument. Next, it is easy to see that the all ones vector is in K. In fact, Hergert,Bauer, and Ganter [1] have already shown this. Note that C is distance invariant, and so the complement of every codeword in C is also in the code.…”

Section: Theorem 1 For All K > 4 There Exists a Nonlinear Hamming Codmentioning

confidence: 87%

Kernels of nonlinear Hamming codes

Phelps

LeVan

1995

Des Codes Crypt

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Abstract. Given a binary code, C, of length n, the kernel of the code is defined to be the set of all vectors which leave the code invariant under translation. Throughout this paper, various properties of kernels will be considered. However, the main idea of this paper is to show the necessary and sufficient conditions for the existence of kernels of all possible sizes for nonlinear perfect binary codes.

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Section: Theorem 1 For All K > 4 There Exists a Nonlinear Hamming Codmentioning

confidence: 87%

Kernels of nonlinear Hamming codes

Phelps

LeVan

1995

Des Codes Crypt

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“…Since 0 ∈ C, K C is a binary linear subcode of C. We denote by κ the dimension of K C . In general, C can be written as a union of cosets of K C , and K C is the largest such linear code for which this is true [1]. Therefore,…”

Section: Representation Of Binary Nonlinear Codesmentioning

confidence: 99%

“…For example, there are binary codes which have a Z 4 -linear or Z 2 Z 4 -linear structure and, therefore, they can also be compactly represented using a quaternary generator matrix [4,12]. In general, binary codes without considering any such structure can be seen as a union of cosets of a binary linear subcode of C [1]. For example, the original Preparata and Kerdock codes are defined like this [16], and new binary nonlinear codes have also been constructed as an union of cosets of a linear code [9,18].…”

mentioning

confidence: 99%

Efficient representation of binary nonlinear codes: constructions and minimum distance computation

Villanueva

Zeng

Pujol

2014

Des. Codes Cryptogr.

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A binary nonlinear code can be represented as a union of cosets of a binary linear subcode. In this paper, the complexity of some algorithms to obtain this representation is analyzed. Moreover, some properties and constructions of new codes from given ones in terms of this representation are described. Algorithms to compute the minimum distance of binary nonlinear codes, based on known algorithms for linear codes, are also established, along with an algorithm to decode such codes. All results are written in such a way that they can be easily transformed into algorithms, and the performance of these algorithms is evaluated.

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“…If so, K(C) is the largest such linear code for which this is true. For p = 2, the kernel was introduced in [2]. …”

Section: The Casementioning

confidence: 99%

Relationships Between CCZ and EA Equivalence Classes and Corresponding Code Invariants

Horadam

Villanueva

2014

Sequences and Their Applications - SETA 2014

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Abstract. The purpose of this paper is to provide a brief survey of CCZ and EA equivalence for functions f : G → N where G and N are finite and N is abelian, and, for the case f : Z m p → Z m p , to investigate two codes derived from f , inspired by these equivalences. In particular we show the dimension of the kernel of each code determines a new invariant of the corresponding equivalence class. We present computational results for p = 2 and small m.

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Algebraic techniques for nonlinear codes

Cited by 61 publications

References 2 publications

Kernels of nonlinear Hamming codes

Kernels of nonlinear Hamming codes

Efficient representation of binary nonlinear codes: constructions and minimum distance computation

Relationships Between CCZ and EA Equivalence Classes and Corresponding Code Invariants

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