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35
The Z_4linearity of Kerdock, Preparata, Goethals, and related codes
, 2001
"... Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by NordstromRobinson, Kerdock, Preparata, Goethals, and DelsarteGoethals. It is shown here that all these codes can be very simply constructed as binary images under the ..."
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Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by NordstromRobinson, Kerdock, Preparata, Goethals, and DelsarteGoethals. It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over ¡ 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 ¡ 4 domain implies that the binary images have dual weight distributions. The Kerdock and ‘Preparata ’ codes are duals over ¡ 4 — and the NordstromRobinson code is selfdual — which explains why their weight distributions are dual to each other. The Kerdock and ‘Preparata ’ codes are ¡ 4analogues of firstorder ReedMuller and extended Hamming codes, respectively. All these codes are extended cyclic codes over ¡ 4, which greatly simplifies encoding and decoding. An algebraic harddecision decoding algorithm is given for the ‘Preparata ’ code and a Hadamardtransform softdecision decoding algorithm for the Kerdock code. Binary first and secondorder ReedMuller codes are also linear over ¡ 4, but extended Hamming codes of length n ≥ 32 and the
A linear construction for certain Kerdock and Preparata codes
 Bull. Amer. Math. Soc
, 1993
"... codes are shown to be linear over Z4, the integers mod 4. The Kerdock and Preparata codes are duals over Z4, and the NordstromRobinson code is selfdual. All these codes are just extended cyclic codes over Z4. This provides a simple definition for these codes and explains why their Hamming weight d ..."
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Cited by 18 (5 self)
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codes are shown to be linear over Z4, the integers mod 4. The Kerdock and Preparata codes are duals over Z4, and the NordstromRobinson code is selfdual. All these codes are just extended cyclic codes over Z4. This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First and secondorder ReedMuller codes are also linear codes over Z4, but Hamming codes in general are not, nor is the Golay code. 1.
Meeting the Welch bound with equality
 in Sequences Their Applications: Proc. SETA ’98
, 1999
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MultipleAccess Interference Reduction for QSCDMA Systems with a Novel Class of Polyphase Sequences
, 2000
"... In this paper, a new class of polyphase sequences for CDMA systems and its generation method are suggested, and the correlation properties of the sequence are investigated. The performance of the sequence in quasisynchronous CDMA (QSCDMA) systems is investigated under frequencyselective, timenon ..."
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Cited by 10 (3 self)
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In this paper, a new class of polyphase sequences for CDMA systems and its generation method are suggested, and the correlation properties of the sequence are investigated. The performance of the sequence in quasisynchronous CDMA (QSCDMA) systems is investigated under frequencyselective, timenonselective, slow Nakagami fading channel with additive white Gaussian noise (AWGN). It is shown that the performance of the QSCDMA systems using the suggested sequence is independent of the number of users and is much better than that using the PN sequence.
Demonstration of a fourchannel WDM/OCDMA system using 255chip 320Gchip/s quarternary phase coding grating
 IEEE Photonics Technology letters
, 2002
"... Abstract—In this letter, we report the fabrication and application of 255chip 320 Gchip/s quaternary phase superstructure fiber Bragg gratings (SSFBGs) for optical code generation and recognition in a fourchannel wavelengthdivisionmultiplexing (WDM)/optical codedivisionmultiplexing (OCDM) exp ..."
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Cited by 7 (1 self)
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Abstract—In this letter, we report the fabrication and application of 255chip 320 Gchip/s quaternary phase superstructure fiber Bragg gratings (SSFBGs) for optical code generation and recognition in a fourchannel wavelengthdivisionmultiplexing (WDM)/optical codedivisionmultiplexing (OCDM) experiment. Individual users of the system operate with different coding schemes, repetition rates, and wavelengths. Our experiments show that a single SSFBG can be used to perform simultaneous optical decoding and wavelength channel selection. Index Terms—Bragg grating, fiberoptic communication, matched filters, optical codedivision multiplexing. I.
Z4Valued Quadratic Forms and Quaternary Sequence Families
, 2009
"... In this paper, Z4valued quadratic forms defined on a vector space over are studied. A classification of such forms is established, distinguishingvalued quadratic forms only by their rank and whether the associated bilinear form is alternating. This result is used to compute the distribution of c ..."
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Cited by 6 (0 self)
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In this paper, Z4valued quadratic forms defined on a vector space over are studied. A classification of such forms is established, distinguishingvalued quadratic forms only by their rank and whether the associated bilinear form is alternating. This result is used to compute the distribution of certain exponential sums, which occur frequently in the analysis of quaternary codes and quaternary sequence sets. The concept is applied as follows. When or is odd, the correlation distribution of family , consisting of quaternary sequences of length , is established. Then, motivated by practical considerations, a subset of family is defined, and the correlation distribution of family is given for odd and even m.
On the Apparent Duality of the Kerdock and Preparata Codes
, 1993
"... . The Kerdock and extended Preparata codes are something of an enigma in coding theory since they are both Hammingdistance invariant and have weight enumerators that are MacWilliams duals just as if they were dual linear codes. In this paper, we explain, by constructing in a natural way a Preparata ..."
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Cited by 6 (2 self)
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. The Kerdock and extended Preparata codes are something of an enigma in coding theory since they are both Hammingdistance invariant and have weight enumerators that are MacWilliams duals just as if they were dual linear codes. In this paper, we explain, by constructing in a natural way a Preparatalike code PL from the Kerdock code K, why the existence of a distanceinvariant code with weight distribution that is the McWilliams transform of that of the Kerdock code is only to be expected. The construction involves quaternary codes over the ring ZZ4 of integers modulo 4. We exhibit a quaternary code Q and its quaternary dual Q ? which, under the Gray mapping, give rise to the Kerdock code K and Preparatalike code PL , respectively. The code PL is identical in weight and distance distribution to the extended Preparata code. The linearity of Q and Q ? ensures that the binary codes K and PL are distance invariant, while their duality as quaternary codes guarantees that K and PL ha...
Translates of Linear Codes Over Z4
, 1997
"... We give a method to compute the complete weight distribution of translates of linear codes over Z4. 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 ..."
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Cited by 4 (0 self)
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We give a method to compute the complete weight distribution of translates of linear codes over Z4. 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.
Algebraic Coding Theory: Recent Developments Related to Z4, Study of Algebraic
 Combinatorics (Proceedings Conference on Algebraic Combinatorics, Kyoto 1993), Research Institute for Mathematical Sciences, Kyoto
, 1995
"... This talk is intended to serve as an introduction to a number of recent papers by various authors that make use of codes over $Z_{4} $ (the integers modulo 4) to greatly simplify the construction of a number of notorious nonlinear binary codes, in particular the codes of NordstromRobinson, Prepara ..."
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Cited by 4 (0 self)
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This talk is intended to serve as an introduction to a number of recent papers by various authors that make use of codes over $Z_{4} $ (the integers modulo 4) to greatly simplify the construction of a number of notorious nonlinear binary codes, in particular the codes of NordstromRobinson, Preparata, Kerdock, Goethals, DelsarteGoethals, Best and Julin. By expanding classical coding theory in this way, allowing alphabets that are rings rather than fields, we are able to give extremely simple descriptions of these codes.