共查询到20条相似文献,搜索用时 10 毫秒
1.
Sergey V. Avgustinovich Olof Heden Faina I. Solov'eva 《Designs, Codes and Cryptography》2004,31(3):313-318
Perfect 1-error correcting codes C in Z
2
n
, where n=2
m–1, are considered. Let
; denote the linear span of the words of C and let the rank of C be the dimension of the vector space
. It is shown that if the rank of C is n–m+2 then C is equivalent to a code given by a construction of Phelps. These codes are, in case of rank n–m+2, described by a Hamming code H and a set of MDS-codes D
h
, h
H, over an alphabet with four symbols. The case of rank n–m+1 is much simpler: Any such code is a Vasil'ev code. 相似文献
2.
Olof Heden 《Designs, Codes and Cryptography》2006,38(1):125-129
A full rank perfect 1-error correcting binary code of length 31 with a kernel of dimension 21 is described. This was the last
open case of the rank-kernel problem of Etzion and Vardy.
AMS Classification: 94B25 相似文献
3.
Mattias Svanström 《Designs, Codes and Cryptography》1999,18(1-3):223-229
We construct a class of perfect ternary constant-weight codes of length 2
r
, weight 2
r
-1 and minimum distance 3. The codes have
codewords. The construction is based on combining cosets of binary Hamming codes. As a special case, for r=2 the construction gives the subcode of the tetracode consisting of its nonzero codewords. By shortening the perfect codes, we get further optimal codes. 相似文献
4.
In this paper, we present three algebraic constructions of authentication codes with secrecy. The first and the third class are optimal. Some of the codes in the second class are optimal, and others in the second class are asymptotically optimal. All authentication codes in the three classes provide perfect secrecy. 相似文献
5.
We consider the space of ternary words of length n and fixed weight w with the usual Hamming distance. A sequence of perfect single error correcting codes in this space is constructed. We prove the nonexistence of such codes with other parameters than those of the sequence. 相似文献
6.
Sergey V. Avgustinovich Olof Heden Faina I. Solov’eva 《Designs, Codes and Cryptography》2006,39(3):317-322
The main result is that to any even integer q in the interval 0 ≤ q ≤ 2n+1-2log(n+1), there are two perfect codes C1 and C2 of length n = 2m − 1, m ≥ 4, such that |C1 ∩ C2| = q. 相似文献
7.
The concept of diameter perfect codes, which seems to be a natural generalization of perfect codes (codesattaining the sphere–packing bound) is introduced. This was motivated by the code–anticode bound of Delsartein distance regular graphs. This bound in conjunction with the recent complete solutions of diametric problems in the Hamming graph q(n) and the Johnson graph J(n,k)gives a sharpening of the sphere–packing bound. Some necessaryconditions for the existence of diameter perfect codes are given.In the Hamming graph all diameter perfect codes over alphabetsof prime power size are characterized. The problem of tilingof the vertex set of J(n,k) with caps (and maximalanticodes) is also examined. 相似文献
8.
Kevin T. Phelps 《Designs, Codes and Cryptography》1999,16(2):179-184
The authors present a 1-error correcting perfect code of length 15 and show that it is not switching equivalent to the Hamming code thereby settling a question of Avgustinovich and Solov'evaas96 相似文献
9.
Existence of Perfect 3-Deletion-Correcting Codes 总被引:1,自引:0,他引:1
A. Mahmoodi 《Designs, Codes and Cryptography》1998,14(1):81-87
Bours [4] recently showed some constructions for perfect 2 and 3-deletion-correcting codes from combinatorial designs. He settled existence of perfect 2-deletion-correcting codes with words of length 4. However, the existence of perfect 3-deletion-correcting codes with words of length 5, or T*(2, 5, v), remained unsettled for v 7, 8 (mod 10) and v = 13, 14, 15, 16. In this paper we provide new constructions for these codes from combinatorial designs, and show that a T*(2, 5, v) exists for all v. 相似文献
10.
Vladimir D. Tonchev 《Designs, Codes and Cryptography》1999,17(1-3):121-128
A new definition for the dimension of a combinatorial t-(v,k,) design over a finite field is proposed. The complementary designs of the hyperplanes in a finite projective or affine geometry, and the finite Desarguesian planes in particular, are characterized as the unique (up to isomorphism) designs with the given parameters and minimum dimension. This generalizes a well-known characterization of the binary hyperplane designs in terms of their minimum 2-rank. The proof utilizes the q-ary analogue of the Hamming code, and a group-theoretic characterization of the classical designs. 相似文献
11.
By a T
*(2, k, v)-code we mean a perfect4-deletion-correcting code of length 6 over an alphabet of size v, which is capable of correcting anycombination of up to 4 deletions and/or insertions of letters that occur in transmission of codewords. Thethird author (DCC Vol. 23, No. 1) presented a combinatorial construction for such codes and prove thata T
*(2, 6, v)-code exists for all positive integers v 3 (mod 5), with 12 possible exceptions of v. In this paper, the notion of a directedgroup divisible quasidesign is introduced and used to show that a T
*(2, 6,v)-code exists for all positive integers v 3 (mod 5), except possiblyfor v {173, 178, 203, 208}. The 12 missing cases for T
*(2,6, v)-codes with v 3 (mod 5) are also provided, thereby the existenceproblem for T
*(2, 6, v)-codes is almost complete. 相似文献
12.
There are exactlytwo non-equivalent [32,11,12]-codes in the binaryReed-Muller code
which contain
and have the weight set {0,12,16,20,32}. Alternatively,the 4-spaces in the projective space
over the vector space
for which all points have rank 4 fall into exactlytwo orbits under the natural action of PGL(5) on
. 相似文献
13.
A trace formula for quasi-cyclic codes over rings of characteristic not coprime with the co-index is derived. The main working
tool is the Generalized Discrete Fourier Transform (GDFT), which in turn relies on the Hasse derivative of polynomials. A
characterization of Type II self-dual quasi-cyclic codes of singly even co-index over finite fields of even characteristic
follows. Implications for generator theory are shown. Explicit expressions for the combinatorial duocubic, duoquintic and
duoseptic constructions in characteristic two over finite fields are given.
AMS Classification: 94B05, 94B15, 11T71
Part of this work was done while the first named author was visiting CNRS-I3S, ESSI, Sophia Antipolis, France. The author
would like to thank the institution for the kind hospitality. The research of the first two authors is partially supported
by MOE-ARF research Grant R-146-000-029-112 and DSTA research Grant R-394-000-011-422. 相似文献
14.
关于完满的Lie超代数 总被引:1,自引:0,他引:1
In this paper, some properties of perfect Lie superalgebras are investigated. We prove that the derivation superalgebra of a centerless perfect Lie superalgebra of arbitrary dimension over a field of arbitrary characteristic is complete and we obtain a necessary and sufficient condition for the holomorph of a centerless perfect Lie superalgebra to be complete. Finally, some properties of perfect restricted Lie superalgebras are given. 相似文献
15.
17.
Christine Bachoc 《Designs, Codes and Cryptography》1999,18(1-3):11-28
We define some new polynomials associated to a linear binary code and a harmonic function of degree k. The case k=0 is the usual weight enumerator of the code. When divided by (xy)
k
, they satisfy a MacWilliams type equality. When applied to certain harmonic functions constructed from Hahn polynomials, they can compute some information on the intersection numbers of the code. As an application, we classify the extremal even formally self-dual codes of length 12. 相似文献
18.
Mihai Ciucu 《Journal of Algebraic Combinatorics》2003,17(3):335-375
In the last decade there have been many results about special families of graphs whose number of perfect matchings is given by perfect or near perfect powers (N. Elkies et al., J. Algebraic Combin.
1 (1992), 111–132; B.-Y. Yang, Ph.D. thesis, Department of Mathematics, MIT, Cambridge, MA, 1991; J. Propp, New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999). In this paper we present an approach that allows proving them in a unified way. We use this approach to prove a conjecture of James Propp stating that the number of tilings of the so-called Aztec dungeon regions is a power (or twice a power) of 13. We also prove a conjecture of Matt Blum stating that the number of perfect matchings of a certain family of subgraphs of the square lattice is a power of 3 or twice a power of 3. In addition we obtain multi-parameter generalizations of previously known results, and new multi-parameter exact enumeration results. We obtain in particular a simple combinatorial proof of Bo-Yin Yang's multivariate generalization of fortresses, a result whose previously known proof was quite complicated, amounting to evaluation of the Kasteleyn matrix by explicit row reduction. We also include a new multivariate exact enumeration of Aztec diamonds, in the spirit of Stanley's multivariate version. 相似文献
19.
20.
Faina I. Soloveva 《Discrete Applied Mathematics》2008,156(9):1488-1498
Some results on perfect codes obtained during the last 6 years are discussed. The main methods to construct perfect codes such as the method of -components and the concatenation approach and their implementations to solve some important problems are analyzed. The solution of the ranks and kernels problem, the lower and upper bounds of the automorphism group order of a perfect code, spectral properties, diameter perfect codes, isometries of perfect codes and codes close to them by close-packed properties are considered. 相似文献