An analogue of the ℤ4-Goethals code in non-primitive length (original) (raw)

On the lifted Zetterberg code

Designs, Codes and Cryptography, 2015

The even-weight subcode of a binary Zetterberg code is a cyclic code with generator polynomial g(x) = (x + 1) p(x), where p(x) is the minimum polynomial over G F(2) of an element of order 2 m + 1 in G F(2 2m) and m is even. This even binary code has parameters [2 m + 1, 2 m − 2m, 6]. The quaternary code obtained by lifting the code to the alphabet Z 4 = {0, 1, 2, 3} is shown to have parameters [2 m + 1, 2 m − 2m, d L ], where d L ≥ 8 denotes the minimum Lee distance. The image of the Gray map of the lifted code is a binary code with parameters (2 m+1 + 2, 2 k , d H), where d H ≥ 8 denotes the minimum Hamming weight and k = 2 m+1 − 4m. For m = 6 these parameters equal the parameters of the best known binary linear code for this length and dimension. Furthermore, a simple algebraic decoding algorithm is presented for these Z 4-codes for all even m. This appears to be the first infinite family of Z 4-codes of length n = 2 m + 1 with d L ≥ 8 having an algebraic decoding algorithm. Keywords Zetterberg code • Cyclic codes • Codes over Z 4 Mathematics Subject Classification 94B15 • 94B35 1 Introduction Codes over Z 4 , the ring of integers modulo 4, have been intensively studied during the last 25 years using the theory of Galois rings, in particular after the ideas developed in the 1990s Communicated by J. Bierbrauer.

The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes

IEEE Transactions on Information Theory, 1994

Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson , Kerdock, Preparata, Goethals, and Delsarte-Goethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z 4 , the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z 4 domain implies that the binary images have dual weight distributions. The Kerdock and 'Preparata' codes are duals over Z 4-and the Nordstrom-Robinson code is self-dual-which explains why their weight distributions are dual to each other. The Kerdock and 'Preparata' codes are Z 4-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z 4 , which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the 'Preparata' code and a Hadamard-transform soft-decision decoding algorithm for the Kerdock code. Binary first-and second-order Reed-Muller codes are also linear over Z 4 , but extended Hamming codes of length n ≥ 32 and the Golay code are not. Using Z 4-linearity, a new family of distance regular graphs are constructed on the cosets of the 'Preparata' code.

A study of cyclic and constacyclic codes over Z4 + uZ4 + vZ4

International Journal of Information and Coding Theory, 2018

In this paper, we study some properties of cyclic and constacyclic codes over the ring R = Z4 + uZ4 + vZ4 where u 2 = v 2 = uv = vu = 0. The generator polynomials and minimal spanning set for cyclic codes over R are determined. Further, (1 + 2u)-constacyclic codes are considered and find the cyclic, quasi-cyclic and permutation equivalent to a QC code over Z4 as the Gray images of (1 + 2u)-constacyclic codes over R.

Quasi-cyclic codes over Z/sub 4/ and some new binary codes

IEEE Transactions on Information Theory, 2002

Recently, (linear) codes over and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study-QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to produces a new binary code, a (92 2 28)-code, which is the best among all binary codes (linear or nonlinear). Moreover, we characterize cyclic codes corresponding to free modules in terms of their generator polynomials.

Codes over Z4 and permutation decoding of linear codes

2017

The Combinatoric, Coding and Security Group (CCSG) is a research group in the Department of Information and Communications Engineering (DEIC) at the Universitat Autònoma de Barcelona (UAB). The research group CCSG has been uninterruptedly working since 1987 in several projects and research activities on Information Theory, Communications, Coding Theory, Source Coding, Cryptography, Electronic Voting, Network Coding, etc. The members of the group have been producing mainly results on optimal coding. Specifically, the research has been focused on uniformly-packed codes; perfect codes in the Hamming space; perfect codes in distance-regular graphs; the classification of optimal codes of a given length; and codes which are close to optimal codes by some properties, for example, Reed-Muller codes, Preparata codes, Kerdock codes and Hadamard codes. Part of the research developed by CCSG deals with codes over Z 4. Some members of CCSG have been developing this new package that expands the current functionality for codes over Z 4 in Magma. Magma is a software package designed to solve computationally hard problems in algebra, number theory, geometry and combinatorics. The latest version of this package for codes over Z 4 and this manual with the description of all developed functions can be downloaded from the web page http://ccsg.uab.cat. For any comment or further information about this package, you can send an e-mail to

Some families of Z4-cyclic codes

Finite Fields and Their Applications, 2004

We introduce and solve several problems on Z 4-cyclic codes.We study the link between Z 4linear cyclic codes and Z 4-cyclic codes (not necessarily linear) obtained by using two binary linear cyclic codes. We use these results to present a family of Z 4-self-dual linear cyclic codes.

Permutation decoding of {\mathbb {Z}}_2{\mathbb {Z}}_4$$ Z 2 Z 4 -linear codes

Designs, Codes and Cryptography, 2014

An alternative permutation decoding method is described which can be used for any binary systematic encoding scheme, regardless whether the code is linear or not. Thus, the method can be applied to some important codes such as Z 2 Z 4-linear codes, which are binary and, in general, nonlinear codes in the usual sense. For this, it is proved that these codes allow a systematic encoding scheme. As particular examples, this permutation decoding method is applied to some Hadamard Z 2 Z 4-linear codes.

There is exactly one Z 2 Z 4-cyclic 1-perfect code

2018

Let C be aZ2Z4-additive code of lengthn > 3. We prove that if the binary Gray image of C, C = Φ(C), is a 1-perfect nonlinear code, then C cannot be aZ2Z4-cyclic code except for one case of length n = 15. Moreover, we give a parity check matrix for this cyclic code. Adding an eve n parity check coordinate to a Z2Z4-additive 1-perfect code gives an extended 1-perfect code. We also prove that any such code cannot be Z2Z4-cyclic.

Four Applications of Z 4 −codes and their GR(4, 2) analogues

S P. Solé,"A quaternary cyclic code, and a family of quadriphase sequences with low correlation properties", Springer Lect. Not. Comp. Sc. 388, 1988. HKCSS Hammons, Kumar, Calderbank, Sloane, Solé, "The Z 4 -linearity of Kerdock Preparata Goethals and related codes"IEEE IT March 94 . BSC Bonnecaze, Solé, Calderbank "Quaternary Construction of Unimodular Lattices" IEEE IT March 95. BRS Bonnecaze, Rains, Solé," 3-Colored 5-Designs and Z 4 -Codes ", J. Statistical Plan. Inf. 2000. Patrick Solé Four Applications of Z 4 −codesand their GR(4, 2) analogues 1988 Quaternary Low Correlation Sequences meeting the Sidelnikov bound (1971)[S] on interference vs length 1994 Explication of the formal duality (MacWilliams transform of weight enumerators) of (nonlinear !) Kerdock and Preparata codes (1972) → Award : Best paper in Information Theory for 1994 [HKCSS] 1995 A new construction of the Leech lattice (1965) [BCS], the building brick of the Conway sporadic simple groups 1999 New 5 − (24, 10, 36) designs supported by the words of the lifted Golay [BRS] (computer find of Harada 96) : proof by invariant theory of weight enumerators Patrick Solé Four Applications of Z 4 −codesand their GR(4, 2) analogues Patrick Solé Four Applications of Z 4 −codesand their GR(4, 2) analogues Patrick Solé Four Applications of Z 4 −codesand their GR(4, 2) analogues Patrick Solé Four Applications of Z 4 −codesand their GR(4, 2) analogues Patrick Solé Four Applications of Z 4 −codesand their GR(4, 2) analogues Patrick Solé Four Applications of Z 4 −codesand their GR(4, 2) analogues Patrick Solé Four Applications of Z 4 −codesand their GR(4, 2) analogues Patrick Solé Four Applications of Z 4 −codesand their GR(4, 2) analogues