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Bonnecaze A. Duursma I.M. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1997,43(4):1218-1230
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 code length 相似文献
3.
Kyeongcheol Yang Helleseth T. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1997,43(6):1832-1842
Hammons et al. (see ibid., vol.40, p.301-19, 1994) showed that, when properly defined, the binary nonlinear Preparata code can be considered as the Gray map of a linear code over Z4, the so called Preparata code over Z4. We consider the rth generalized Hamming weight dr(m) of the Preparata code of length 2m over Z4. For any m⩾3, dr(m) is exactly determined for r=0.5, 1, 1.5, 2, 2.5 and 3.0. For a composite m, we give an upper bound on dr(m) using the lifting technique. For m=3, 4, 5, 6 and 8, the weight hierarchy is completely determined. In the case of m=7, the weight hierarchy is completely determined except for d4(7) 相似文献
4.
Helleseth T. Hove B. Kyeongcheol Yang 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1999,45(4):1255-1258
This article contains results on the generalized Hamming weights (GHW) for the Goethals and Preparata codes over Z4. We give an upper bound on the rth generalized Hamming weights dr(m,j) for the Goethals code Gm(j) of length 2m over Z 4, when m is odd. We also determine d3.5(m,j) exactly. The upper bound is shown to be tight up to r=3.5. Furthermore, we determine the rth generalized Hamming weight dr(m) for the Preparata code of length 2m over Z4 when r=3.5 and r=4 相似文献
5.
Helleseth T. Zinoviev V. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》2001,47(5):1758-1772
We study the coset weight distributions of two well-known families of codes: the three-error-correcting binary Z4-linear Goethals codes of length N=2m+1, m⩾3 odd, and the Z4 -linear Goethals codes over Z4 of length n=N/2=2m . The hard case is the weight distributions of cosets of weight 4. To know the weight distribution of the coset of weight 4 we have to know the number of codewords of weight 4 in such a coset. Altogether, there are nine different types of cosets of weight 4. For six cases, we give the exact expressions for the number of codewords of weight 4, and for three other cases, we give such expressions in terms of Kloosterman sums 相似文献
6.
Calderbank A.R. McGuire G. Kumar V.P. Helleseth T. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1996,42(1):217-226
Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock (1972) and Preparata (1968) codes that can be very simply constructed as binary images, under the Gray map, of linear codes over Z4 that are defined by means of parity checks involving Galois rings. This paper describes how Fourier transforms on Galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes. These methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of Z 4. Linear codes with length 2m (m, odd) and size 2(2m+1-5m-2). The Gray image of the code of length 32 is the best (64, 237) code that is presently known. This paper also determines the exact minimum Lee distance of the linear codes over Z4 that are obtained from the extended binary two- and three-error-correcting BCH codes by Hensel lifting. The Gray image of the Hensel lift of the three-error-correcting BCH code of length 32 is the best (64, 232) code that is presently known. This code also determines an extremal 32-dimensional even unimodular lattice 相似文献
7.
Encheva S.B. Jensen H.E. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1996,42(4):1216-1222
We give necessary and sufficient conditions for a binary linear code to be Z4-linear. Especially we treat optimal, binary linear codes and determine all such codes with minimum weight less or equal to six which are Z4-linear 相似文献
8.
Piret P.M. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1995,41(3):815-818
Let S(8) denote the set of the eight admissible signals of an 8PSK communication system. The alphabet S(8) is endowed with the structure of Z8, the set of integers taken modulo 8, and codes are defined to be Z8-submodules of Z8n. Three cyclic codes over Z8 are then constructed. Their length is equal to 6, 8, and 7, and they, respectively, contain 64, 64, and 512 codewords. The square of their Euclidean minimum distance is equal to 8, 16-4√2 and 10-2√2, respectively. The size of the codes of length 6 and 7 can be doubled while the Euclidean minimum distance remains the same 相似文献
9.
Tanabe K. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》2000,46(1):48-53
The Assmus-Mattson theorem is a method to find designs in linear codes over a finite field. The purpose of this paper is to give an analog of this theorem for Z4-codes by using the harmonic weight enumerator introduced by Bachoc. This theorem can find some 5-designs in the lifted Golay code over Z4 which were discovered previously by other methods 相似文献
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Carlet C. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1998,44(4):1543-1547
We introduce a generalization to Z2k of the Gray map and generalized versions of Kerdock and Delsarte-Goethals codes 相似文献
11.
Wolfmann J. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》2001,47(5):1773-1779
We determine all linear cyclic codes over Z 4 of odd length whose Gray images are linear codes (or, equivalently, whose Nechaev-Gray (1989) image are linear cyclic codes or are linear cyclic codes) 相似文献
12.
Ranto K. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》2000,46(6):2193-2197
The Z4-linear Goethals-like code of length 2m has 22m+1-3m-2 codewords and minimum Lee distance 8 for any odd integer m⩾3. We present an algebraic decoding algorithm for all Z4-linear Goethals-like codes Ck introduced by Helleseth et al.(1995, 1996). We use Dickson polynomials and their properties to solve the syndrome equations 相似文献
13.
Aydin N. Ray-Chaudhuri D.K. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》2002,48(7):2065-2069
Previously, (linear) codes over Z 4 and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study Z 4-QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to Z 4 produces a new binary code, a (92, 224, 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 相似文献
14.
Johannesson R. Wittenmark E. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1998,44(4):1602-1604
For rate R=1/2 convolutional codes with 16 states there exists a gap between Heller's (1968) upper bound on the free distance and its optimal value. This article reports on the construction of 16-state, binary, rate R=2/4 nonlinear trellis and convolutional codes having d free=8; a free distance that meets the Heller upper bound. The nonlinear trellis code is constructed from a 16-state, rate R=1/2 convolutional code over Z4 using the Gray map to obtain a binary code. Both convolutional codes are obtained by computer search. Systematic feedback encoders for both codes are potential candidates for use in combination with iterative decoding. Regarded as modulation codes for 4-PSK, these codes have free squared Euclidean distance dE, free2=16 相似文献
15.
Helleseth T. Kumar P.V. Moreno O. Shanbhag A.G. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1996,42(4):1212-1216
An upper hound for Weil-type exponential sums over Galois rings was derived by Kumar, Helleseth, and Calderbank (see ibid., vol.41, no.3, p.456, 1995). This bound leads directly to an estimate for the minimum distance of Z4-linear trace codes. An improved minimum-distance estimate is presented. First, McEliece's result on the divisibility of the weights of binary cyclic codes is extended to Z4 trace codes. The divisibility result is then combined with the techniques of Serre (1983) and of Moreno and Moreno (see ibid., vol.40, no.11, p.1101, 1994) to derive the improved minimum-distance estimate. The improved estimate is tight for the Kerdock code as well as for the Delsarte-Goethals codes 相似文献
16.
Punctured convolutional codes of rates k1/n and k2 /n are applied to |u|u+v construction, and then a superimposed code of rate (k1+k2)/(2n) is constructed. A suboptimal decoding procedure is presented for the superimposed codes, and it reduces the decoding complexity as compared with maximum likelihood decoding for the known convolutional codes 相似文献
17.
Carlet C. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1995,41(5):1487-1494
Recently the notion on binary codes called Z4-linearity was introduced. This notion explains why Kerdock codes and Delsarte-Goethals codes admit formal duals in spite of their nonlinearity. The “Z4-duals” of these codes (called “Preparata” and “Goethals” codes) are new nonlinear codes which admit simpler decoding algorithms than the previously known formal duals (the generalized Preparata and Goethals codes). We prove, by using the notion of exact weight enumerator, that the relationship between any Z4-linear code and its Z4 -dual is stronger than the standard formal duality and we deduce the weight enumerators of related generalized codes 相似文献
18.
Udaya P. Siddiqi M.U. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1996,42(1):206-216
New families of biphase sequences of size 2r-1+1, r being a positive integer, are derived from families of interleaved maximal-length sequences over Z4 of period 2(2r-1). These sequences have applications in code-division spread-spectrum multiuser communication systems. The families satisfy the Sidelnikov bound with equality on &thetas;max, which denotes the maximum magnitude of the periodic cross-correlation and out-of-phase autocorrelation values. One of the families satisfies the Welch bound on &thetas;max with equality. The linear complexity and the period of all sequences are equal to r(r+3)/2 and 2(2 r-1), respectively, with an exception of the single m-sequence which has linear complexity r and period 2r-1. Sequence imbalance and correlation distributions are also computed 相似文献
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Hammons A.R. Jr. Kumar P.V. Calderbank A.R. Sloane N.J.A. Sole P. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1994,40(2):301-319
Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson (1967), Kerdock (1972), Preparata (1968), Goethals (1974), and Delsarte-Goethals (1975). 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 I(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 相似文献