-
Previous Article
Comparison analysis of Ding's RLWE-based key exchange protocol and NewHope variants
- AMC Home
- This Issue
- Next Article
Self-dual additive $ \mathbb{F}_4 $-codes of lengths up to 40 represented by circulant graphs
Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980–8579, Japan |
In this paper, we consider additive circulant graph codes which are self-dual additive $ \mathbb{F}_4 $-codes. We classify all additive circulant graph codes of length $ n = 30, 31 $ and $ 34 \le n \le 40 $ having the largest minimum weight. We also classify bordered circulant graph codes of lengths up to 40 having the largest minimum weight.
References:
[1] |
B. Alspach and T. D. Parsons,
Isomorphism of circulant graphs and digraphs, Discrete Math., 25 (1979), 97-108.
doi: 10.1016/0012-365X(79)90011-6. |
[2] |
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. |
[3] |
A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane,
Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
doi: 10.1109/18.681315. |
[4] |
S. Cichacz and D. Froncek,
Distance magic circulant graphs, Discrete Math., 339 (2016), 84-94.
doi: 10.1016/j.disc.2015.07.002. |
[5] |
L. E. Danielsen and M. G. Parker,
On the classification of all self-dual additive codes over GF(4) of length up to $12$, J. Combin. Theory Ser. A, 113 (2006), 1351-1367.
doi: 10.1016/j.jcta.2005.12.004. |
[6] |
L. E. Danielsen and M. G. Parker,
Directed graph representation of half-rate additive codes over GF(4), Des. Codes Cryptogr., 59 (2011), 119-130.
doi: 10.1007/s10623-010-9469-6. |
[7] |
B. Elspas and J. Turner,
Graphs with circulant adjacency matrices, J. Combinatorial Theory, 9 (1970), 297-307.
doi: 10.1016/S0021-9800(70)80068-0. |
[8] |
M. Grassl and M. Harada,
New self-dual additive $\mathbb{F}_4$-codes constructed from circulant graphs, Discrete Math., 340 (2017), 399-403.
doi: 10.1016/j.disc.2016.08.023. |
[9] |
F. J. MacWilliams, A. M. Odlyzko, N. J. A. Sloane and H. N. Ward,
Self-dual codes over $GF(4)$, J. Combin. Theory Ser. A, 25 (1978), 288-318.
doi: 10.1016/0097-3165(78)90021-3. |
[10] |
Z. Varbanov,
Additive circulant graph codes over GF(4), Math. Maced., 6 (2008), 73-79.
|
[11] |
Z. Varbanov,
T. Todorov and M. Hristova, A method for constructing DNA codes from additive self-dual codes over GF(4), ROMAI J., 10 (2014), 203-211.
|
show all references
References:
[1] |
B. Alspach and T. D. Parsons,
Isomorphism of circulant graphs and digraphs, Discrete Math., 25 (1979), 97-108.
doi: 10.1016/0012-365X(79)90011-6. |
[2] |
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. |
[3] |
A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane,
Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
doi: 10.1109/18.681315. |
[4] |
S. Cichacz and D. Froncek,
Distance magic circulant graphs, Discrete Math., 339 (2016), 84-94.
doi: 10.1016/j.disc.2015.07.002. |
[5] |
L. E. Danielsen and M. G. Parker,
On the classification of all self-dual additive codes over GF(4) of length up to $12$, J. Combin. Theory Ser. A, 113 (2006), 1351-1367.
doi: 10.1016/j.jcta.2005.12.004. |
[6] |
L. E. Danielsen and M. G. Parker,
Directed graph representation of half-rate additive codes over GF(4), Des. Codes Cryptogr., 59 (2011), 119-130.
doi: 10.1007/s10623-010-9469-6. |
[7] |
B. Elspas and J. Turner,
Graphs with circulant adjacency matrices, J. Combinatorial Theory, 9 (1970), 297-307.
doi: 10.1016/S0021-9800(70)80068-0. |
[8] |
M. Grassl and M. Harada,
New self-dual additive $\mathbb{F}_4$-codes constructed from circulant graphs, Discrete Math., 340 (2017), 399-403.
doi: 10.1016/j.disc.2016.08.023. |
[9] |
F. J. MacWilliams, A. M. Odlyzko, N. J. A. Sloane and H. N. Ward,
Self-dual codes over $GF(4)$, J. Combin. Theory Ser. A, 25 (1978), 288-318.
doi: 10.1016/0097-3165(78)90021-3. |
[10] |
Z. Varbanov,
Additive circulant graph codes over GF(4), Math. Maced., 6 (2008), 73-79.
|
[11] |
Z. Varbanov,
T. Todorov and M. Hristova, A method for constructing DNA codes from additive self-dual codes over GF(4), ROMAI J., 10 (2014), 203-211.
|
Ref. | Ref. | ||||||||
- | - | [10] | |||||||
1 | [10] | ||||||||
- | - | [10] | |||||||
2 | [10] | ||||||||
- | - | [10] | |||||||
1 | [10] | ||||||||
- | - | [10] | |||||||
1 | [10] | ||||||||
- | - | [10] | |||||||
5 | |||||||||
- | - | ||||||||
1 | [10] | ||||||||
- | [10] | - | [10] | ||||||
3 | [10] | ||||||||
- | [10] | - | |||||||
5 | [10] | ||||||||
- | [10] | - | |||||||
36 | [10] | ||||||||
- | [10] | - | |||||||
2 | [10] |
Ref. | Ref. | ||||||||
- | - | [10] | |||||||
1 | [10] | ||||||||
- | - | [10] | |||||||
2 | [10] | ||||||||
- | - | [10] | |||||||
1 | [10] | ||||||||
- | - | [10] | |||||||
1 | [10] | ||||||||
- | - | [10] | |||||||
5 | |||||||||
- | - | ||||||||
1 | [10] | ||||||||
- | [10] | - | [10] | ||||||
3 | [10] | ||||||||
- | [10] | - | |||||||
5 | [10] | ||||||||
- | [10] | - | |||||||
36 | [10] | ||||||||
- | [10] | - | |||||||
2 | [10] |
Code | ||||||||
Code | ||||||||
Ref. | Ref. | ||||||
- | - | - | - | ||||
[6] | [6] | ||||||
[6] | |||||||
[6] | |||||||
[6] | |||||||
[6] | |||||||
[6] | |||||||
[6] | |||||||
[6] | |||||||
[6] |
Ref. | Ref. | ||||||
- | - | - | - | ||||
[6] | [6] | ||||||
[6] | |||||||
[6] | |||||||
[6] | |||||||
[6] | |||||||
[6] | |||||||
[6] | |||||||
[6] | |||||||
[6] |
Code | ||||||||
Code | ||||||||
[1] |
Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027 |
[2] |
Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261 |
[3] |
W. Cary Huffman. Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order. Advances in Mathematics of Communications, 2007, 1 (3) : 357-398. doi: 10.3934/amc.2007.1.357 |
[4] |
Padmapani Seneviratne, Martianus Frederic Ezerman. New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2021073 |
[5] |
Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45 |
[6] |
Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032 |
[7] |
Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503 |
[8] |
T. Aaron Gulliver, Masaaki Harada, Hiroki Miyabayashi. Double circulant and quasi-twisted self-dual codes over $\mathbb F_5$ and $\mathbb F_7$. Advances in Mathematics of Communications, 2007, 1 (2) : 223-238. doi: 10.3934/amc.2007.1.223 |
[9] |
María Chara, Ricardo A. Podestá, Ricardo Toledano. The conorm code of an AG-code. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2021018 |
[10] |
Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275 |
[11] |
W. Cary Huffman. Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order. Advances in Mathematics of Communications, 2013, 7 (1) : 57-90. doi: 10.3934/amc.2013.7.57 |
[12] |
Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1 |
[13] |
Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003 |
[14] |
Lars Eirik Danielsen. Graph-based classification of self-dual additive codes over finite fields. Advances in Mathematics of Communications, 2009, 3 (4) : 329-348. doi: 10.3934/amc.2009.3.329 |
[15] |
Joaquim Borges, Steven T. Dougherty, Cristina Fernández-Córdoba. Characterization and constructions of self-dual codes over $\mathbb Z_2\times \mathbb Z_4$. Advances in Mathematics of Communications, 2012, 6 (3) : 287-303. doi: 10.3934/amc.2012.6.287 |
[16] |
Delphine Boucher. Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$. Advances in Mathematics of Communications, 2016, 10 (4) : 765-795. doi: 10.3934/amc.2016040 |
[17] |
Andrea Seidl, Stefan Wrzaczek. Opening the source code: The threat of forking. Journal of Dynamics and Games, 2022 doi: 10.3934/jdg.2022010 |
[18] |
Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011 |
[19] |
Masaaki Harada. Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights. Advances in Mathematics of Communications, 2016, 10 (4) : 695-706. doi: 10.3934/amc.2016035 |
[20] |
Jianying Fang. 5-SEEDs from the lifted Golay code of length 24 over Z4. Advances in Mathematics of Communications, 2017, 11 (1) : 259-266. doi: 10.3934/amc.2017017 |
2020 Impact Factor: 0.935
Tools
Metrics
Other articles
by authors
[Back to Top]