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1.
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
2.
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
3.
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
4.
5.
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
6.
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
7.
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
8.
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
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
10.
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
11.
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
12.
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
13.
《IEEE transactions on information theory / Professional Technical Group on Information Theory》1983,29(3):345-348
Ifm is odd andsigma /in Aut GF(2^{m}) is such thatx rightarrow x^{sigma^{2}-1} is1-1 , there is a[2^{m+1}-1,2^{m+l}-2m-2] nonlinear binary codeP(sigma) having minimum distance 5. All the codesP(sigma) have the same distance and weight enumerators as the usual Preparata codes (which rise asP(sigma) whenx^{sigma}=x^{2}) . It is shown thatP(sigma) andP(tau) are equivalent if and only iftau=sigma^{pm 1} , andAut P(sigma) is determined. 相似文献
14.
研究了环F4+uF4与域F4上的线性码,利用环F4+uF4上码C的Gray重量wG,Gray距离d G和(F4+uF4)n到F4 2n的Gray映射φ,证明了环F4+uF4上线性码C及其对偶码的Gray像φ(C)为F4上的线性码和对偶且dH G(φ(C))dG(C)。同时,给出了F4+uF4上循环码C的Gray像φ(C)为F4上的2-拟循环码。 相似文献
15.
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
16.
Benedetto S. Garello R. Mondin M. Montorsi G. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》1993,39(6):1773-1798
The theory of geometrically uniform trellis codes is applied to the case of multidimensional PSK (phase shift keying) constellations. The symmetry group of an L×MPSK (M-ary PSK) constellation is completely characterized. Conditions for rotational invariance of geometrically uniform partitions of a signal constellation are given. Through suitable algorithms, geometrically uniform partitions of L×MPSK (M=4,8,16 and L=1,2,3,4) constellations are found, which present good characteristics in terms of the set of distances at a given partition level, the maximum obtainable rotational invariance, and the isomorphism of the quotient group associated with the partition. These partitions are used as starting points in a search for good geometrically uniform trellis codes based on binary convolutional codes 相似文献
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18.
Betsumiya K. Gulliver T.A. Harada M. Munemasa A. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》2001,47(6):2242-2248
Previously, Type II codes over F4 have been introduced as Euclidean self-dual codes with the property that all Lee weights are divisible by four. In this paper, a number of properties of Type II codes are presented. We construct several extremal Type II codes and a number of extremal Type I codes. It is also shown that there are seven Type II codes of length 12, up to permutation equivalence 相似文献
19.
有限环Z4上码字广度的性质及其递归算法 总被引:1,自引:0,他引:1
研究码及码字的结构是编码理论的一个重要研究方向.该文定义了环Z4上码字的一种数学特征,即码字的广度.研究了码字广度的一些性质,给出了计算Z4环上码字广度的两种递归算法,并对Z4环上的码字广度与深度之间的关系进行了初步的讨论. 相似文献
20.
Gulliver T.A. 《IEEE transactions on information theory / Professional Technical Group on Information Theory》2000,46(1):271-274
Optimal double circulant self-dual codes over F4 have been found for each length n⩽40. For lengths n⩽14, 20, 22, 24, 28, and 30, these codes are optimal self-dual codes. For length 26, the code attains the highest known minimum weight. For n⩾32, the codes presented provide the highest known minimum weights. The [36,18,12] self-dual code improves the lower bound on the highest minimum weight for a [36,18] linear code 相似文献