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# Self-dual additive $\mathbb{F}_4$-codes of lengths up to 40 represented by circulant graphs

• 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.

Mathematics Subject Classification: Primary: 94B25; Secondary: 94B05.

 Citation:

• Table 1.  Additive circulant graph codes

 $n$ $d_{\rm max}^{\rm A}(n)$ ${\rm num}^{\rm A}_{\rm I}(n)$ ${\rm num}^{\rm A}_{\rm II}(n)$ 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] $40$ $12$ $0$ $213$

Table 2.  Weight distributions of $C(\Gamma_{31}^{(i)})$ $(i = 1, \ldots, 5)$

 Code $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $C(\Gamma_{31}^{(1)})$ $0$ $1$ $15$ $2000988$ $21$ $215937072$ $27$ $111769756$ $10$ $1209$ $16$ $6017193$ $22$ $294597774$ $28$ $47879469$ $11$ $7564$ $17$ $15948384$ $23$ $345959256$ $29$ $14851976$ $12$ $34441$ $18$ $37215066$ $24$ $345825894$ $30$ $2973179$ $13$ $154504$ $19$ $76416984$ $25$ $290407008$ $31$ $288332$ $14$ $593991$ $20$ $137479482$ $26$ $201124125$ $C(\Gamma_{31}^{(2)})$ $0$ $1$ $15$ $1990944$ $21$ $215937072$ $27$ $111779800$ $10$ $1209$ $16$ $6043977$ $22$ $294504030$ $28$ $47886165$ $11$ $7192$ $17$ $15966240$ $23$ $345974880$ $29$ $14849000$ $12$ $35185$ $18$ $37152570$ $24$ $345888390$ $30$ $2972435$ $13$ $157480$ $19$ $76401360$ $25$ $290389152$ $31$ $288704$ $14$ $587295$ $20$ $137573226$ $26$ $201097341$ $C(\Gamma_{31}^{(3)})$ $0$ $1$ $15$ $2004336$ $21$ $215937072$ $27$ $111766408$ $10$ $1209$ $16$ $6008265$ $22$ $294629022$ $28$ $47877237$ $11$ $7688$ $17$ $15942432$ $23$ $345954048$ $29$ $14852968$ $12$ $34193$ $18$ $37235898$ $24$ $345805062$ $30$ $2973427$ $13$ $153512$ $19$ $76422192$ $25$ $290412960$ $31$ $288208$ $14$ $596223$ $20$ $137448234$ $26$ $201133053$ $C(\Gamma_{31}^{(4)})$ $0$ $1$ $15$ $2007374$ $21$ $216042968$ $27$ $111761262$ $10$ $1395$ $16$ $6022649$ $22$ $294539618$ $28$ $47920265$ $11$ $6758$ $17$ $15937968$ $23$ $345831660$ $29$ $14845156$ $12$ $35557$ $18$ $37226350$ $24$ $345942454$ $30$ $2966421$ $13$ $155620$ $19$ $76386604$ $25$ $290475952$ $31$ $290502$ $14$ $587481$ $20$ $137474274$ $26$ $201025359$ $C(\Gamma_{31}^{(5)})$ $0$ $1$ $15$ $2004584$ $21$ $216058592$ $27$ $111758472$ $10$ $1333$ $16$ $6030089$ $22$ $294526598$ $28$ $47920885$ $11$ $6696$ $17$ $15945408$ $23$ $345818640$ $29$ $14845776$ $12$ $36177$ $18$ $37213330$ $24$ $345949894$ $30$ $2966359$ $13$ $156240$ $19$ $76373584$ $25$ $290483392$ $31$ $290440$ $14$ $584691$ $20$ $137489898$ $26$ $201022569$

Table 3.  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] $40$ $12$ $27$

Table 4.  Weight distributions of $\overline{C}(\Gamma_{n-1})$ $(n = 29, 30, 31, 37)$

 Code $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $i$ $A_i$ $\overline{C}(\Gamma_{28})$ $0$ $1$ $14$ $960696$ $20$ $70176246$ $26$ $13926402$ $9$ $196$ $15$ $1404096$ $21$ $80787000$ $27$ $7162400$ $10$ $4130$ $16$ $6819393$ $22$ $90249720$ $28$ $879975$ $11$ $4704$ $17$ $9807336$ $23$ $88070976$ $29$ $169548$ $12$ $69027$ $18$ $29368108$ $24$ $55981758$ $13$ $127932$ $19$ $37819264$ $25$ $43082004$ $\overline{C}(\Gamma_{29})$ $0$ $1$ $16$ $12038625$ $22$ $341403660$ $28$ $18581895$ $12$ $118755$ $18$ $61752600$ $24$ $312800670$ $30$ $378018$ $14$ $1151010$ $20$ $195945750$ $26$ $129570840$ $\overline{C}(\Gamma_{30})$ $0$ $1$ $15$ $1296630$ $21$ $195080760$ $27$ $129747510$ $10$ $1931$ $16$ $7888953$ $22$ $310437330$ $28$ $38102265$ $11$ $3534$ $17$ $11648880$ $23$ $342154620$ $29$ $18540660$ $12$ $51285$ $18$ $45631390$ $24$ $334681590$ $30$ $2109261$ $13$ $85620$ $19$ $62449660$ $25$ $312351600$ $31$ $382350$ $14$ $830385$ $20$ $156683394$ $26$ $177324039$ $\overline{C}(\Gamma_{36})$ $0$ $1$ $17$ $9143640$ $24$ $8309464632$ $31$ $11671999680$ $11$ $360$ $18$ $67292720$ $25$ $10288765008$ $32$ $4976806725$ $12$ $11520$ $19$ $102308360$ $26$ $16810527456$ $33$ $3176508888$ $13$ $19068$ $20$ $516207384$ $27$ $18804288888$ $34$ $730497264$ $14$ $318384$ $21$ $741731364$ $28$ $20519937680$ $35$ $305859096$ $15$ $533376$ $22$ $2581317216$ $29$ $20147052420$ $36$ $28393304$ $16$ $5746818$ $23$ $3466908864$ $30$ $14172955632$ $37$ $4357724$
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