# American Institute of Mathematical Sciences

May  2019, 13(2): 213-220. doi: 10.3934/amc.2019014

## 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

Received  January 2017 Published  February 2019

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.

Citation: Ken Saito. Self-dual additive $\mathbb{F}_4$-codes of lengths up to 40 represented by circulant graphs. Advances in Mathematics of Communications, 2019, 13 (2) : 213-220. doi: 10.3934/amc.2019014
##### References:
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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] S. Cichacz and D. Froncek, Distance magic circulant graphs, Discrete Math., 339 (2016), 84-94.  doi: 10.1016/j.disc.2015.07.002.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] Z. Varbanov, Additive circulant graph codes over GF(4), Math. Maced., 6 (2008), 73-79.   Google Scholar [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.   Google Scholar
 $n$ $d_{\rm max}^{\rm A}(n)$ ${\rm num}^{\rm A}_{\rm I}(n)$ ${\rm num}^{\rm A}_{\rm II}(n)$ Ref. $n$ $d_{\rm max}^{\rm A}(n)$ ${\rm num}^{\rm A}_{\rm I}(n)$ ${\rm num}^{\rm A}_{\rm II}(n)$ Ref. $1$ $1$ $1$ - $21$ $7$ $11$ - [10] $2$ $2$ $0$ 1 $22$ $8$ $0$ $14$ [10] $3$ $2$ $1$ - $23$ $8$ $2$ - [10] $4$ $2$ $1$ 2 $24$ $8$ $5$ $46$ [10] $5$ $3$ $1$ - $25$ $8$ $31$ - [10] $6$ $4$ $0$ 1 $26$ $8$ $49$ $161$ [10] $7$ $3$ $1$ - $27$ $8$ $140$ - [10] $8$ $4$ $0$ 1 $28$ $10$ $0$ $1$ [10] $9$ $4$ $1$ - $29$ $11$ $1$ - [10] $10$ $4$ $3$ 5 $30$ $12$ $0$ $1$ $11$ $4$ $2$ - $31$ $10$ $5$ - $12$ $6$ $0$ 1 $32$ $10$ $2$ $106$ [10] $13$ $5$ $2$ - [10] $33$ $10$ $76$ - [10] $14$ $6$ $0$ 3 [10] $34$ $10$ $115$ $851$ $15$ $6$ $2$ - [10] $35$ $10$ $595$ - $16$ $6$ $1$ 5 [10] $36$ $11$ $1$ $0$ $17$ $7$ $1$ - [10] $37$ $11$ $17$ - $18$ $6$ $16$ 36 [10] $38$ $12$ $0$ $22$ $19$ $7$ $4$ - [10] $39$ $11$ $276$ - $20$ $8$ $0$ 2 [10] $40$ $12$ $0$ $213$
 $n$ $d_{\rm max}^{\rm A}(n)$ ${\rm num}^{\rm A}_{\rm I}(n)$ ${\rm num}^{\rm A}_{\rm II}(n)$ Ref. $n$ $d_{\rm max}^{\rm A}(n)$ ${\rm num}^{\rm A}_{\rm I}(n)$ ${\rm num}^{\rm A}_{\rm II}(n)$ Ref. $1$ $1$ $1$ - $21$ $7$ $11$ - [10] $2$ $2$ $0$ 1 $22$ $8$ $0$ $14$ [10] $3$ $2$ $1$ - $23$ $8$ $2$ - [10] $4$ $2$ $1$ 2 $24$ $8$ $5$ $46$ [10] $5$ $3$ $1$ - $25$ $8$ $31$ - [10] $6$ $4$ $0$ 1 $26$ $8$ $49$ $161$ [10] $7$ $3$ $1$ - $27$ $8$ $140$ - [10] $8$ $4$ $0$ 1 $28$ $10$ $0$ $1$ [10] $9$ $4$ $1$ - $29$ $11$ $1$ - [10] $10$ $4$ $3$ 5 $30$ $12$ $0$ $1$ $11$ $4$ $2$ - $31$ $10$ $5$ - $12$ $6$ $0$ 1 $32$ $10$ $2$ $106$ [10] $13$ $5$ $2$ - [10] $33$ $10$ $76$ - [10] $14$ $6$ $0$ 3 [10] $34$ $10$ $115$ $851$ $15$ $6$ $2$ - [10] $35$ $10$ $595$ - $16$ $6$ $1$ 5 [10] $36$ $11$ $1$ $0$ $17$ $7$ $1$ - [10] $37$ $11$ $17$ - $18$ $6$ $16$ 36 [10] $38$ $12$ $0$ $22$ $19$ $7$ $4$ - [10] $39$ $11$ $276$ - $20$ $8$ $0$ 2 [10] $40$ $12$ $0$ $213$
Weight distributions of $C(\Gamma_{31}^{(i)})$ $(i = 1, \ldots, 5)$
 Code $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $C(\Gamma_{31}^{(1)})$ $0$ $1$ $15$ $2000988$ $21$ $215937072$ $27$ $111769756$ $10$ $1209$ $16$ $6017193$ $22$ $294597774$ $28$ $47879469$ $11$ $7564$ $17$ $15948384$ $23$ $345959256$ $29$ $14851976$ $12$ $34441$ $18$ $37215066$ $24$ $345825894$ $30$ $2973179$ $13$ $154504$ $19$ $76416984$ $25$ $290407008$ $31$ $288332$ $14$ $593991$ $20$ $137479482$ $26$ $201124125$ $C(\Gamma_{31}^{(2)})$ $0$ $1$ $15$ $1990944$ $21$ $215937072$ $27$ $111779800$ $10$ $1209$ $16$ $6043977$ $22$ $294504030$ $28$ $47886165$ $11$ $7192$ $17$ $15966240$ $23$ $345974880$ $29$ $14849000$ $12$ $35185$ $18$ $37152570$ $24$ $345888390$ $30$ $2972435$ $13$ $157480$ $19$ $76401360$ $25$ $290389152$ $31$ $288704$ $14$ $587295$ $20$ $137573226$ $26$ $201097341$ $C(\Gamma_{31}^{(3)})$ $0$ $1$ $15$ $2004336$ $21$ $215937072$ $27$ $111766408$ $10$ $1209$ $16$ $6008265$ $22$ $294629022$ $28$ $47877237$ $11$ $7688$ $17$ $15942432$ $23$ $345954048$ $29$ $14852968$ $12$ $34193$ $18$ $37235898$ $24$ $345805062$ $30$ $2973427$ $13$ $153512$ $19$ $76422192$ $25$ $290412960$ $31$ $288208$ $14$ $596223$ $20$ $137448234$ $26$ $201133053$ $C(\Gamma_{31}^{(4)})$ $0$ $1$ $15$ $2007374$ $21$ $216042968$ $27$ $111761262$ $10$ $1395$ $16$ $6022649$ $22$ $294539618$ $28$ $47920265$ $11$ $6758$ $17$ $15937968$ $23$ $345831660$ $29$ $14845156$ $12$ $35557$ $18$ $37226350$ $24$ $345942454$ $30$ $2966421$ $13$ $155620$ $19$ $76386604$ $25$ $290475952$ $31$ $290502$ $14$ $587481$ $20$ $137474274$ $26$ $201025359$ $C(\Gamma_{31}^{(5)})$ $0$ $1$ $15$ $2004584$ $21$ $216058592$ $27$ $111758472$ $10$ $1333$ $16$ $6030089$ $22$ $294526598$ $28$ $47920885$ $11$ $6696$ $17$ $15945408$ $23$ $345818640$ $29$ $14845776$ $12$ $36177$ $18$ $37213330$ $24$ $345949894$ $30$ $2966359$ $13$ $156240$ $19$ $76373584$ $25$ $290483392$ $31$ $290440$ $14$ $584691$ $20$ $137489898$ $26$ $201022569$
 Code $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $C(\Gamma_{31}^{(1)})$ $0$ $1$ $15$ $2000988$ $21$ $215937072$ $27$ $111769756$ $10$ $1209$ $16$ $6017193$ $22$ $294597774$ $28$ $47879469$ $11$ $7564$ $17$ $15948384$ $23$ $345959256$ $29$ $14851976$ $12$ $34441$ $18$ $37215066$ $24$ $345825894$ $30$ $2973179$ $13$ $154504$ $19$ $76416984$ $25$ $290407008$ $31$ $288332$ $14$ $593991$ $20$ $137479482$ $26$ $201124125$ $C(\Gamma_{31}^{(2)})$ $0$ $1$ $15$ $1990944$ $21$ $215937072$ $27$ $111779800$ $10$ $1209$ $16$ $6043977$ $22$ $294504030$ $28$ $47886165$ $11$ $7192$ $17$ $15966240$ $23$ $345974880$ $29$ $14849000$ $12$ $35185$ $18$ $37152570$ $24$ $345888390$ $30$ $2972435$ $13$ $157480$ $19$ $76401360$ $25$ $290389152$ $31$ $288704$ $14$ $587295$ $20$ $137573226$ $26$ $201097341$ $C(\Gamma_{31}^{(3)})$ $0$ $1$ $15$ $2004336$ $21$ $215937072$ $27$ $111766408$ $10$ $1209$ $16$ $6008265$ $22$ $294629022$ $28$ $47877237$ $11$ $7688$ $17$ $15942432$ $23$ $345954048$ $29$ $14852968$ $12$ $34193$ $18$ $37235898$ $24$ $345805062$ $30$ $2973427$ $13$ $153512$ $19$ $76422192$ $25$ $290412960$ $31$ $288208$ $14$ $596223$ $20$ $137448234$ $26$ $201133053$ $C(\Gamma_{31}^{(4)})$ $0$ $1$ $15$ $2007374$ $21$ $216042968$ $27$ $111761262$ $10$ $1395$ $16$ $6022649$ $22$ $294539618$ $28$ $47920265$ $11$ $6758$ $17$ $15937968$ $23$ $345831660$ $29$ $14845156$ $12$ $35557$ $18$ $37226350$ $24$ $345942454$ $30$ $2966421$ $13$ $155620$ $19$ $76386604$ $25$ $290475952$ $31$ $290502$ $14$ $587481$ $20$ $137474274$ $26$ $201025359$ $C(\Gamma_{31}^{(5)})$ $0$ $1$ $15$ $2004584$ $21$ $216058592$ $27$ $111758472$ $10$ $1333$ $16$ $6030089$ $22$ $294526598$ $28$ $47920885$ $11$ $6696$ $17$ $15945408$ $23$ $345818640$ $29$ $14845776$ $12$ $36177$ $18$ $37213330$ $24$ $345949894$ $30$ $2966359$ $13$ $156240$ $19$ $76373584$ $25$ $290483392$ $31$ $290440$ $14$ $584691$ $20$ $137489898$ $26$ $201022569$
Bordered circulant graph codes
 $n$ $d_{\rm max}^{\rm B}(n)$ ${\rm num}^{\rm B}(n)$ Ref. $n$ $d_{\rm max}^{\rm B}(n)$ ${\rm num}^{\rm B}(n)$ Ref. - - - - $21$ $6$ $34$ $2$ $2$ $1$ [6] $22$ $8$ $3$ [6] $3$ $2$ $1$ [6] $23$ $7$ $20$ $4$ $2$ $1$ $24$ $8$ $11$ $5$ $2$ $2$ $25$ $8$ $18$ $6$ $4$ $1$ [6] $26$ $8$ $14$ $7$ $3$ $1$ $27$ $8$ $70$ $8$ $4$ $1$ [6] $28$ $8$ $102$ $9$ $4$ $1$ [6] $29$ $9$ $1$ $10$ $4$ $1$ $30$ $12$ $1$ $11$ $4$ $3$ $31$ $10$ $1$ $12$ $4$ $1$ $32$ $10$ $41$ $13$ $5$ $1$ $33$ $10$ $31$ $14$ $6$ $2$ [6] $34$ $10$ $368$ $15$ $6$ $1$ [6] $35$ $10$ $381$ $16$ $6$ $3$ $36$ $10$ $249$ $17$ $6$ $4$ $37$ $11$ $1$ $18$ $8$ $1$ [6] $38$ $12$ $4$ $19$ $6$ $25$ $39$ $11$ $22$ $20$ $8$ $2$ [6] $40$ $12$ $27$
 $n$ $d_{\rm max}^{\rm B}(n)$ ${\rm num}^{\rm B}(n)$ Ref. $n$ $d_{\rm max}^{\rm B}(n)$ ${\rm num}^{\rm B}(n)$ Ref. - - - - $21$ $6$ $34$ $2$ $2$ $1$ [6] $22$ $8$ $3$ [6] $3$ $2$ $1$ [6] $23$ $7$ $20$ $4$ $2$ $1$ $24$ $8$ $11$ $5$ $2$ $2$ $25$ $8$ $18$ $6$ $4$ $1$ [6] $26$ $8$ $14$ $7$ $3$ $1$ $27$ $8$ $70$ $8$ $4$ $1$ [6] $28$ $8$ $102$ $9$ $4$ $1$ [6] $29$ $9$ $1$ $10$ $4$ $1$ $30$ $12$ $1$ $11$ $4$ $3$ $31$ $10$ $1$ $12$ $4$ $1$ $32$ $10$ $41$ $13$ $5$ $1$ $33$ $10$ $31$ $14$ $6$ $2$ [6] $34$ $10$ $368$ $15$ $6$ $1$ [6] $35$ $10$ $381$ $16$ $6$ $3$ $36$ $10$ $249$ $17$ $6$ $4$ $37$ $11$ $1$ $18$ $8$ $1$ [6] $38$ $12$ $4$ $19$ $6$ $25$ $39$ $11$ $22$ $20$ $8$ $2$ [6] $40$ $12$ $27$
Weight distributions of $\overline{C}(\Gamma_{n-1})$ $(n = 29, 30, 31, 37)$
 Code $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $\overline{C}(\Gamma_{28})$ $0$ $1$ $14$ $960696$ $20$ $70176246$ $26$ $13926402$ $9$ $196$ $15$ $1404096$ $21$ $80787000$ $27$ $7162400$ $10$ $4130$ $16$ $6819393$ $22$ $90249720$ $28$ $879975$ $11$ $4704$ $17$ $9807336$ $23$ $88070976$ $29$ $169548$ $12$ $69027$ $18$ $29368108$ $24$ $55981758$ $13$ $127932$ $19$ $37819264$ $25$ $43082004$ $\overline{C}(\Gamma_{29})$ $0$ $1$ $16$ $12038625$ $22$ $341403660$ $28$ $18581895$ $12$ $118755$ $18$ $61752600$ $24$ $312800670$ $30$ $378018$ $14$ $1151010$ $20$ $195945750$ $26$ $129570840$ $\overline{C}(\Gamma_{30})$ $0$ $1$ $15$ $1296630$ $21$ $195080760$ $27$ $129747510$ $10$ $1931$ $16$ $7888953$ $22$ $310437330$ $28$ $38102265$ $11$ $3534$ $17$ $11648880$ $23$ $342154620$ $29$ $18540660$ $12$ $51285$ $18$ $45631390$ $24$ $334681590$ $30$ $2109261$ $13$ $85620$ $19$ $62449660$ $25$ $312351600$ $31$ $382350$ $14$ $830385$ $20$ $156683394$ $26$ $177324039$ $\overline{C}(\Gamma_{36})$ $0$ $1$ $17$ $9143640$ $24$ $8309464632$ $31$ $11671999680$ $11$ $360$ $18$ $67292720$ $25$ $10288765008$ $32$ $4976806725$ $12$ $11520$ $19$ $102308360$ $26$ $16810527456$ $33$ $3176508888$ $13$ $19068$ $20$ $516207384$ $27$ $18804288888$ $34$ $730497264$ $14$ $318384$ $21$ $741731364$ $28$ $20519937680$ $35$ $305859096$ $15$ $533376$ $22$ $2581317216$ $29$ $20147052420$ $36$ $28393304$ $16$ $5746818$ $23$ $3466908864$ $30$ $14172955632$ $37$ $4357724$
 Code $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $\overline{C}(\Gamma_{28})$ $0$ $1$ $14$ $960696$ $20$ $70176246$ $26$ $13926402$ $9$ $196$ $15$ $1404096$ $21$ $80787000$ $27$ $7162400$ $10$ $4130$ $16$ $6819393$ $22$ $90249720$ $28$ $879975$ $11$ $4704$ $17$ $9807336$ $23$ $88070976$ $29$ $169548$ $12$ $69027$ $18$ $29368108$ $24$ $55981758$ $13$ $127932$ $19$ $37819264$ $25$ $43082004$ $\overline{C}(\Gamma_{29})$ $0$ $1$ $16$ $12038625$ $22$ $341403660$ $28$ $18581895$ $12$ $118755$ $18$ $61752600$ $24$ $312800670$ $30$ $378018$ $14$ $1151010$ $20$ $195945750$ $26$ $129570840$ $\overline{C}(\Gamma_{30})$ $0$ $1$ $15$ $1296630$ $21$ $195080760$ $27$ $129747510$ $10$ $1931$ $16$ $7888953$ $22$ $310437330$ $28$ $38102265$ $11$ $3534$ $17$ $11648880$ $23$ $342154620$ $29$ $18540660$ $12$ $51285$ $18$ $45631390$ $24$ $334681590$ $30$ $2109261$ $13$ $85620$ $19$ $62449660$ $25$ $312351600$ $31$ $382350$ $14$ $830385$ $20$ $156683394$ $26$ $177324039$ $\overline{C}(\Gamma_{36})$ $0$ $1$ $17$ $9143640$ $24$ $8309464632$ $31$ $11671999680$ $11$ $360$ $18$ $67292720$ $25$ $10288765008$ $32$ $4976806725$ $12$ $11520$ $19$ $102308360$ $26$ $16810527456$ $33$ $3176508888$ $13$ $19068$ $20$ $516207384$ $27$ $18804288888$ $34$ $730497264$ $14$ $318384$ $21$ $741731364$ $28$ $20519937680$ $35$ $305859096$ $15$ $533376$ $22$ $2581317216$ $29$ $20147052420$ $36$ $28393304$ $16$ $5746818$ $23$ $3466908864$ $30$ $14172955632$ $37$ $4357724$
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