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The connection between quadratic bent-negabent functions and the kerdock code

APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2019年第5期30卷 387-401页

作者： Stanica, Pantelimon Mandal, Bimal Maitra, Subhamoy Naval Postgrad Sch Dept Appl Math Monterey CA 93943 USA Indian Stat Inst RC Bose Ctr Secur & Cryptol Kolkata India Indian Stat Inst Appl Stat Unit Kolkata India

In this paper we prove that all bent functions in the kerdock code, except for the coset of the symmetric quadratic bent function, are bent-negabent. In this direction, we characterize the set of quadratic bent-negabent functions and show some results connecting quadratic bent-negabent functions and the kerdock code. Further, we note that there are bent-negabent preserving nonsingular transformations outside the well known class of orthogonal ones that might provide additional functions in the bent-negabent set. This is the first time we could identify non-orthogonal (nonsingular) linear transformations that preserve bent-negabent property for a special subset.

关键词： Boolean function Bent function Negabent function kerdock code

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On the (2,1)-separating weight of the kerdock code

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IEEE TRANSACTIONS ON INFORMATION THEORY 2004年第12期50卷 3312-3315页

作者： Helleseth, T Schaathun, HG Univ Bergen Selmer Ctr Dept Informat N-5020 Bergen Norway

Separating codes find applications in many fields including automata theory and digital fingerprinting. It is known that the kerdock code of sufficient order is (2, 1)- and (2, 2) -separating, but the separating weight is only known by a lower bound due to Sagalovich. In this correspondence, we prove that the lower bound on the (2, 1)-separating weight is met with equality.

关键词： fingerprinting kerdock code linear codes over Z(4) separating systems

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On components of the kerdock codes and the dual of the BCH code C1,3

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DISCRETE MATHEMATICS 2020年第2期343卷

作者： Mogilnykh, I. Yu. Solov'eva, F. I. Sobolev Inst Math Novosibirsk Russia Novosibirsk State Univ Novosibirsk Russia

In the paper we investigate the structure of i-components of two classes of codes: the kerdock codes and the duals of the linear uniformly packed codes with parameters of the primitive double-error-correcting BCH code. We prove that for any admissible length the punctured kerdock code consists of two i-components and the duals of the linear uniformly packed codes with parameters of the primitive double-error-correcting BCH code is an i-component for any i. We give an alternative proof for the fact presented by De Caen and van Dam in 1999 that the restrictions of the Hamming scheme to the doubly shortened kerdock codes are association schemes. (C) 2019 Published by Elsevier B.V.

关键词： kerdock code BCH code Uniformly packed code Association scheme i-component t-design

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Enumeration of kerdock codes of length 64

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DESIGNS codeS AND CRYPTOGRAPHY 2015年第2-3期77卷 357-363页

作者： Phelps, Kevin Auburn Univ Dept Math & Stat Auburn AL 36849 USA

An algorithm for enumerating kerdock codes is discussed and used to show that the kerdock code of length 64 is unique up to equivalence.

关键词： kerdock code Reed-Muller code Z(4)-code Bent function

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Distance regularity of kerdock codes

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SIBERIAN MATHEMATICAL JOURNAL 2008年第3期49卷 539-548页

作者： Solov'eva, F. I. Tokareva, N. N. Sobolev Inst Math Novosibirsk Russia

A code is called distance regular, if for every two codewords x, y and integers i, j the number of codewords z such that d(x, z) = i and d(y, z) = j, with d the Hamming distance, does not depend on the choice of x, y and depends only on d(x, y) and i, j. Using some properties of the discrete Fourier transform we give a new combinatorial proof of the distance regularity of an arbitrary kerdock code. We also calculate the parameters of the distance regularity of a kerdock code.

关键词： distance regular code kerdock code Reed-Muller code discrete Fourier transform bent function distance regular graph association scheme

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Translates of linear codes over Z(4)

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IEEE TRANSACTIONS ON INFORMATION THEORY 1997年第4期43卷 1218-1230页

作者： Bonnecaze, A Duursma, IM 13S LAB F-06903 SOPHIA ANTIPOLISFRANCE LMD EQUIPE ATI F-13288 MARSEILLE 9FRANCE

We give a method to compute the complete weight distribution of translates of linear codes over Z(4). The method follows known ideas that have already been used successfully by others for Hamming weight distributions. For the particular case of quaternary Preparata codes, we obtain that the number of distinct complete weights for the dual Preparata codes and the number of distinct complete coset weight enumerators for the Preparata codes are both equal to ten, independent of the codelength.

关键词： complete weight enumerator coset weight enumerator kerdock code Preparata code quaternary code

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Split weight enumerators for the preparata codes with applications to designs

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DESIGNS codeS AND CRYPTOGRAPHY 1999年第1-3期18卷 103-124页

作者： Duursma, I Helleseth, T Rong, CM Yang, KC AT&T Labs Res Murray Hill NJ 07974 USA Univ Bergen Dept Informat N-5020 Bergen Norway Pohang Univ Sci & Technol Dept Elect & Elect Engn Pohang 790784 Kyungbuk South Korea

For quaternary Preparata and kerdock codes of length N=2(m), m odd, we prove that the split complete weight enumerator for a coordinate partition into 3 and N-3 coordinates is independent of the chosen partition. The result implies that the words of a given complete weight in either a Preparata code or kerdock code define a 3-design.

关键词： Preparata code kerdock code 3-design split weight enumerator

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On the existence and construction of good codes with low peak-to-average power ratios

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IEEE TRANSACTIONS ON INFORMATION THEORY 2000年第6期46卷 1974-1987页

作者： Paterson, KG Tarokh, V Hewlett Packard Labs Bristol BS34 8QZ Avon England AT&T Labs Res Florham Pk NJ 07932 USA

The first lower bound on the peak-to-average power ratio (PAPR) of a constant energy code of a given length n, minimum Euclidean distance and rate is established. Conversely, using a nonconstructive Varshamov-Gilbert style argument yields a lower bound on the achievable rate of a code of a given length, minimum Euclidean distance and maximum PAPR, The derivation of these bounds relies on a geometrical analysis of the PAPR of such a code. Further analysis shows that there exist asymptotically good codes whose PAPR is at most 8 log n, These bounds motivate the explicit construction of error-correcting codes with low PAPR, Bounds for exponential sums over Galois fields and rings are applied to obtain an upper bound of order (log n)(2) on the PAPRs of a constructive class of codes, the trace codes. This class includes the binary simplex code, duals of binary, primitive Bose-Chaudhuri-Hocquenghem (BCH) codes and a variety of their nonbinary analogs. Some open problems are identified.

关键词： bounds Delsarte-Goethals code dual BCH code exponential sum finite field Galois ring Gilbert kerdock code Lagrange interpolation multicarrier PAPR PMEPR PMPR power OFDM simplex code Varshamov

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On the trellis representation of the Delsarte-Goethals codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 1998年第4期44卷 1547-1554页

作者： Shany, Y Reuven, I Be'ery, Y Tel Aviv Univ Dept Elect Syst Engn IL-69978 Ramat Aviv Israel

In this correspondence, the trellis representation of the kerdock and Delsarte-Goethals codes is addressed. It is shown that the states of a trellis representation of DG(m, delta) under any bit-order are either strict-sense nonmerging or strict-sense nonexpanding, except, maybe, at indices within the code's distance set. For delta greater than or equal to 3 and for m greater than or equal to 6, the slate complexity, s(max) [DG(m, delta)], is found. For all values of m and delta, a formula for the number of states and branches of the biproper trellis diagram of DG(m, delta) is given for some of the indices, and upper and lower bounds are given for the remaining indices. The formula and the bounds refer to the Delsarte-Goethals codes when arranged in the standard bit-order.

关键词： biproper trellis Delsarte-Goethals code kerdock code rectangular codes trellis complexity

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Two "Dual" families of nearly-linear codes over Ζ_p,p odd

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APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING 2001年第4期11卷 313-329页

作者： van Asch, B van Tilborg, HCA Eindhoven Univ Technol Dept Math & Comp Sci NL-5600 MB Eindhoven Netherlands

Since the paper by Hammons e.a. [1], various authors have shown an enormous interest in linear codes over the ring Z(4). A special weight function on Z(4) was introduced and by means of the so called Gray map phi : Z(4) --> Z(2)(2) a relation was established between linear codes over Z(4) and certain interesting non-linear binary codes of even length. Here, we shall generalize these notions to codes over Z(p)(2) where p is an arbitrary prime. To this end, a new weight function will be proposed for Z(p2). Further, properties of linear codes over Z(p2) will be discussed and the mapping phi will be generalized to an isometry between Z(p2) and Z(p)(p) resp. between Z(p2)(n) and Z(p)(pn). Some properties of Galois rings over Z(q) will be described and two dual families of linear codes of length n = p(m) - 1, gcd(m, p) = 1, over Z(q) will be constructed. Taking q = p(2), their images under the new mapping can be viewed as a generalization of the binary kerdock and the Preparata code, although they miss some of their nice combinatorial properties.

关键词： p(2)-ary codes Gray map Galois rings kerdock code Preparata code

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