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1.
In this note, we give the complete classification of binary formally self-dual even codes of lengths 10, 12, 14 and 16. There are exactly fourteen, 29, 99 and 914 inequivalent such codes of lengths 10, 12, 14 and 16, respectively. This completes the classification of formally self-dual even codes of lengths up to 16. The first example of formally self-dual even code with a trivial automorphism group is also found. This shows that 16 is the smallest length for which there is a formally self-dual even code with a trivial automorphism group. 相似文献
2.
J. E. Fields P. Gaborit W. C. Huffman V. Pless 《Designs, Codes and Cryptography》1999,18(1-3):125-148
Bachoc bachoc has recently introduced harmonic polynomials for binary codes. Computing these for extremal even formally self-dual codes of length 12, she found intersection numbers for such codes and showed that there are exactly three inequivalent [12,6,4] even formally self-dual codes, exactly one of which is self-dual. We prove a new theorem which gives a generator matrix for formally self-dual codes. Using the Bachoc polynomials we can obtain the intersection numbers for extremal even formally self-dual codes of length 14. These same numbers can also be obtained from the generator matrix. We show that there are precisely ten inequivalent [14,7,4] even formally self-dual codes, only one of which is self-dual. 相似文献
3.
Masaaki Harada 《Journal of Combinatorial Theory, Series A》2009,116(5):1063-1072
Ternary self-dual codes have been classified for lengths up to 20. At length 24, a classification of only extremal self-dual codes is known. In this paper, we give a complete classification of ternary self-dual codes of length 24 using the classification of 24-dimensional odd unimodular lattices. 相似文献
4.
We study odd and even \(\mathbb{Z }_2\mathbb{Z }_4\) formally self-dual codes. The images of these codes are binary codes whose weight enumerators are that of a formally self-dual code but may not be linear. Three constructions are given for formally self-dual codes and existence theorems are given for codes of each type defined in the paper. 相似文献
5.
Nikolay Yankov 《Designs, Codes and Cryptography》2013,69(2):151-159
We classify up to equivalence all optimal binary self-dual [52, 26, 10] codes having an automorphism of order 3 with 10 fixed points. We achieve this using a method for constructing self-dual codes via an automorphism of odd prime order. We study also codes with an automorphism of order 3 with 4 fixed points. Some of the constructed codes have new values β = 8, 9, and 12 for the parameter in their weight enumerator. 相似文献
6.
We show that (n, 2 n ) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n, 2 n ) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally self-dual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual codes. 相似文献
7.
T. Aaron Gulliver Masaaki Harada Takuji Nishimura Patric R. J. Östergård 《Designs, Codes and Cryptography》2005,37(3):465-471
The weight enumerator of a formally self-dual even code is obtained by the Gleason theorem. Recently, Kim and Pless gave some
restrictions on the possible weight enumerators of near-extremal formally self-dual even codes of length divisible by eight.
In this paper, the weight enumerators for which there is a near-extremal formally self-dual even code are completely determined
for lengths 24 and 32, by constructing new near-extremal formally self-dual codes. We also give a classification of near-
extremal double circulant codes of lengths 24 and 32.
Communicated by: P. Fitzpatrick 相似文献
8.
Most of the research on formally self-dual (f.s.d.) codes has been developed for binary f.s.d. even codes, but only limited research has been done for binary f.s.d.
odd codes. In this article we complete the classification of binary f.s.d. odd codes of lengths up to 14. We also classify
optimal binary f.s.d. odd codes of length 18 and 24, so our result completes the classification of binary optimal f.s.d. odd
codes of lengths up to 26. For this classification we first find a relation between binary f.s.d. odd codes and binary f.s.d.
even codes, and then we use this relation and the known classification results on binary f.s.d. even codes. We also classify
(possibly) optimal binary double circulant f.s.d. odd codes of lengths up to 40. 相似文献
9.
《Finite Fields and Their Applications》2002,8(1):34-51
All (Hermitian) self-dual [24, 12, 8] quaternary codes which have a non-trivial automorphism of order 3 are obtained up to equivalence. There exist exactly 205 inequivalent such codes. The codes under consideration are optimal, self-dual, and have the highest possible minimum distance for this length. 相似文献
10.
A method for constructing binary self-dual codes having an automorphism of order p
2 for an odd prime p is presented in (S. Bouyuklieva et al. IEEE. Trans. Inform. Theory, 51, 3678–3686, 2005). Using this method, we investigate
the optimal self-dual codes of lengths 60 ≤ n ≤ 66 having an automorphism of order 9 with six 9-cycles, t cycles of length 3 and f fixed points. We classify all self-dual [60,30,12] and [62,31,12] codes possessing such an automorphism, and we construct
many doubly-even [64,32,12] and singly-even [66,33,12] codes. Some of the constructed codes of lengths 62 and 66 are with
weight enumerators for which the existence of codes was not known until now.
相似文献
11.
In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a classification of extremal odd unimodular lattices in dimension 40 with shadows having 80 vectors of norm 2 through their relationships with extremal doubly even self-dual codes of length 40. 相似文献
12.
Hyun Jin Kim 《Designs, Codes and Cryptography》2012,63(1):43-57
We classify the extremal self-dual codes of lengths 38 or 40 having an automorphism of order 3 with six independent 3-cycles,
10 independent 3-cycles, or 12 independent 3-cycles. In this way we complete the classification of binary extremal self-dual
codes of length up to 48 having automorphism of odd prime order. 相似文献
13.
14.
Yasemin Cengellenmis Abdullah Dertli S. T. Dougherty 《Designs, Codes and Cryptography》2014,72(3):559-580
Codes over an infinite family of rings which are an extension of the binary field are defined. Two Gray maps to the binary field are attached and are shown to be conjugate. Euclidean and Hermitian self-dual codes are related to binary self-dual and formally self-dual codes, giving a construction of formally self-dual codes from a collection of arbitrary binary codes. We relate codes over these rings to complex lattices. A Singleton bound is proved for these codes with respect to the Lee weight. The structure of cyclic codes and their Gray image is studied. Infinite families of self-dual and formally self-dual quasi-cyclic codes are constructed from these codes. 相似文献
15.
Lengths 22 and 30 are so far the only open cases in the classification of extremal formally self-dual even codes. In this
paper, a classification of the extremal formally self-dual even codes of length 22 is given. There are 41520 such codes.A
variety of properties of these codes are investigated. In particular, new 2-(22, 6, 5) designs are constructed from the codes.
Received: February 9, 2000 相似文献
16.
Formally self-dual even codes have recently been studied. Double circulant even codes are a family of such codes and almost all known extremal formally self-dual even codes are of this form. In this paper, we classify all extremal double circulant formally self-dual even codes which are not self-dual. We also investigate the existence of near-extremal formally self-dual even codes. 相似文献
17.
It is a well-known fact that if C is an [n,k,d] formally self-dual even code with n>30, then d?2[n/8]. A formally self-dual (f.s.d.) even code with d=2[n/8] is called near-extremal. Kim and Pless [A note on formally self-dual even codes of length divisible by 8, Finite Fields Appl., available online 13 October 2005.] conjecture that there does not exist a near-extremal f.s.d. (not Type II) code of length n?48 with 8|n. In this paper, we prove that if n?72 and 8|n, then there is no near-extremal f.s.d. even code. This result comes from the negative coefficients of weight enumerators. In addition, we introduce shadow transform in near-extremal f.s.d. even codes. Using this we present some results about the nonexistence of near-extremal f.s.d. even codes with n=48,64. 相似文献
18.
A. Melakhessou K. Guenda T. A. Gulliver M. Shi P. Solé 《Journal of Applied Mathematics and Computing》2018,57(1-2):375-391
In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R. 相似文献
19.
20.
Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over $\mathbb{F}_{2}$ , $\mathbb{F}_{3}$ , $\mathbb{F}_{4}$ , $\mathbb{F}_{5}$ , and $\mathbb{F}_{7}$ . In particular, we find a new ternary self-dual [76,38,18] code and easily rediscover optimal binary self-dual codes with parameters [66,33,12], [68,34,12], [86,43,16], and [88,44,16] as well as a formally self-dual binary [82,41,14] code. 相似文献