|
||||
11 | 24 | 10 | ||
11 | 28 | 11 | ||
13 | 28 | 12 | ||
13 | 32 | 13 | ||
13 | 36 | 14 | ||
13 | 40 | 15 | ||
17 | 24 | 11 | ||
17 | 28 | 12 | ||
17 | 32 | 13 | ||
17 | 36 | 14 | ||
17 | 40 | 15 | ||
19 | 24 | 12 | ||
19 | 28 | 12 | ||
23 | 28 | 14 | ||
23 | 32 | 13 | ||
23 | 36 | 15 | ||
23 | 40 | 16 |
We construct self-dual codes over F11, F13, F17, F19 and F23 which improve the previously known lower bounds on the largest minimum weights. In particular, the largest possible minimum weight among self-dual [n,n/2] codes over Fp is determined for (p,n)=(19,24) and (23,28).
Citation: |
Table 1.
New four-negacirculant self-dual codes
|
||||
11 | 24 | 10 | ||
11 | 28 | 11 | ||
13 | 28 | 12 | ||
13 | 32 | 13 | ||
13 | 36 | 14 | ||
13 | 40 | 15 | ||
17 | 24 | 11 | ||
17 | 28 | 12 | ||
17 | 32 | 13 | ||
17 | 36 | 14 | ||
17 | 40 | 15 | ||
19 | 24 | 12 | ||
19 | 28 | 12 | ||
23 | 28 | 14 | ||
23 | 32 | 13 | ||
23 | 36 | 15 | ||
23 | 40 | 16 |
Table 2.
New double circulant self-dual codes
|
|||
13 | 26 | 11 | |
13 | 30 | 12 | |
13 | 38 | 14 |
Table 3.
New quasi-twisted self-dual codes
|
|||
13 | 34 | 13 | |
17 | 26 | 11 |
Table 4.
Weight enumerators of self-dual
|
|
0 | |
12 | |
13 | |
14 | |
15 | |
16 | |
17 | |
18 | |
19 | |
20 | |
21 | |
22 | |
23 | |
24 |
Table 5.
Weight enumerators of self-dual
|
|
0 | 1 |
14 | |
15 | |
16 | |
17 | |
18 | |
19 | |
20 | |
21 | |
22 | |
23 | |
24 | |
25 | |
26 | |
27 | |
28 |
Table 6.
|
11 | 13 | 17 | 19 | 23 |
14 | 8 | 7–8 | |||
16 | 8 | 8 | 8–9 | 8–9 | 9 |
18 | 9 | 10 | |||
20 | 10 | 10 | 10 | 11 | 10–11 |
22 | 10–11 | 10–11 | |||
24 | 10–12 | 10–12 | 11–12 | 12 | 13 |
26 | 11–13 | 11–13 | |||
28 | 11–14 | 12–14 | 12–14 | 12–14 | 14 |
30 | 12–15 | 12–15 | |||
32 | 12–16 | 13–16 | 13–16 | 14–16 | 13–16 |
34 | 13–17 | 13–17 | |||
36 | 13–18 | 14–18 | 14-18 | 14–18 | 15–18 |
38 | 14–19 | 14–19 | |||
40 | 14–20 | 15–20 | 15–20 | 15–20 | 16–20 |
[1] | K. Betsumiya, S. Georgiou, T. A. Gulliver, M. Harada and C. Koukouvinos, On self-dual codes over some prime fields, Discrete Math., 262 (2003), 37-58. doi: 10.1016/S0012-365X(02)00520-4. |
[2] | M. A. De Boer, Almost MDS codes, Des. Codes Cryptogr., 9 (1996), 143-155. doi: 10.1007/BF00124590. |
[3] | W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125. |
[4] | W.-H. Choi and J.-L. Kim, Self-dual codes, symmetric matrices, and eigenvectors, IEEE Access, 9 (2021), 104294-104303. |
[5] | W. H. Choi and J. L. Kim, An improved upper bound on self-dual codes over finite fields GF(11), GF(19), and GF(23), to appear, Des. Codes Cryptogr.. |
[6] | P. Gaborit, Quadratic double circulant codes over fields, J. Comb. Theory Ser. A, 97 (2002), 85-107. doi: 10.1006/jcta.2001.3198. |
[7] | S. D. Georgiou and E. Lappas, Self-dual codes from circulant matrices, Des. Codes Cryptogr., 64 (2012), 129-141. doi: 10.1007/s10623-011-9510-4. |
[8] | M. Grassl and T. A. Gulliver, On self-dual MDS codes, ISIT 2008, Toronto, Canada, (2008), 1954-1957. |
[9] | M. Grassl and T. A. Gulliver, On circulant self-dual codes over small fields, Des. Codes Cryptogr., 52 (2009), 57-81. doi: 10.1007/s10623-009-9267-1. |
[10] | T. A. Gulliver and M. Harada, Double circulant self-dual codes over GF(5), Ars Combin., 56 (2000), 3-13. |
[11] | T. A. Gulliver and M. Harada, New nonbinary self-dual codes, IEEE Trans. Inform. Theory, 54 (2008), 415-417. doi: 10.1109/TIT.2007.911265. |
[12] | T. A. Gulliver, M. Harada and H. Miyabayashi, Double circulant and quasi-twisted self-dual codes over F5 and F7, Adv. Math. Commun., 1 (2007), 223-238. doi: 10.3934/amc.2007.1.223. |
[13] | T. A. Gulliver, J.-L. Kim and Y. Lee, New MDS or near-MDS self-dual codes, IEEE Trans. Inform. Theory, 54 (2008), 4354-4360. doi: 10.1109/TIT.2008.928297. |
[14] | S. Han and J.-L. Kim, Computational results of duadic double circulant codes, J. Appl. Math. Comput., 40 (2012), 33-43. doi: 10.1007/s12190-012-0543-2. |
[15] | M. Harada, W. Holzmann, H. Kharaghani and M. Khorvash, Extremal ternary self-dual codes constructed from negacirculant matrices, Graphs Combin., 23 (2007), 401-417. doi: 10.1007/s00373-007-0731-2. |
[16] | F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 18 (1972), 794-805. doi: 10.1109/tit.1972.1054898. |
[17] | M. Shi, L. Sok, P. Solé and S. Çalkavur, Self-dual codes and orthogonal matrices over large finite fields, Finite Fields Appl., 54 (2018), 297-314. doi: 10.1016/j.ffa.2018.08.011. |
[18] | L. Sok, Explicit constructions of MDS self-dual codes, IEEE Trans. Inform. Theory, 66 (2020), 3603-3615. doi: 10.1109/TIT.2019.2954877. |