Факултет по математика и информатика, том ХVІ С, 014 SOME NEW SELF-DUAL [96, 48, 16] CODES WITH AN AUTOMORPHISM OF ORDER 15 NIKOLAY I. YANKOV ABSTRACT: A new method fo constuctng bnay self-dual codes wth an automophsm of ode pq fo p q was developed n [1]. We use ths method to constuct new doubly-even self-dual [96, 48, 16] codes havng an automophsm of ode 15 wth 6 cycles of length 15 and two cycles of length 3. Moe than 100000 new such codes ae obtan. We found exactly 19 dffeent values of the paamete n the weght dstbuton of these codes of whch 11 ae new. KEYWORDS: automophsms, constucton, self-dual codes 1. Intoducton A lnea [ n, k] code C s a k-dmensonal subspace of the vecto space GF( q ) n, whee GF( q ) s the fnte feld of q elements. The elements of C ae called codewods and the (Hammng) weght of a codewod s the numbe of ts nonzeo coodnate postons. The mnmum weght d of C s the smallest weght among all nonzeo codewods of C, and C s called a [ n, k, d] code. A matx whch ows fom a bass of C s called the geneato matx of ths code. The weght enumeato W (y) of a code C s gven by W n ( y ) = A y = 0 whee A, s the numbe of codewods of n n weght n C. Let ( u, v) : Fq Fq Fq be an nne poduct n the lnea space F. The dual code of C s C n q n = { u F : ( u, v) = 0 fo all v C}. The dual code q C s a Ths wok s suppoted by Shumen Unvesty unde Poect RD-08-34/1.03.014-99 -
Факултет по математика и информатика, том ХVІ С, 014 lnea [ n, n k] code. We call the code C self-othogonal f C C. If C = C then the code C s temed self-dual. Two bnay codes ae equvalent f one can be obtaned fom the othe by a pemutaton of coodnates. The pemutaton σ S n s an automophsm of C, f C = σ (C). The set of all automophsms of C foms a goup, called the automophsm goup Aut (C) of C. Fo two dffeent pmes p < q we say that an automophsm σ of ode pq s of type pq (, c tp, tq, f) f t has c cycles of length pq, t p cycles of length p, t q cycles of length q and f fxed ponts n ts decomposton nto dsont cycles. A self-dual code C s doubly-even f all codewods of C have a weght dvsble by fou and sngly-even f thee s at least one codewod of weght conguent modulo 4. Rans n [1] poved that the mnmum dstance d of a bnay self-dual [ nkd,, ] code satsfes the followng bound: d 4 n/ 4 + 4, f n / ( mod 4 ), d 4 n/ 4 + 6, f n ( mod 4 ). Codes achevng ths bound ae called extemal. If n s a multple of 4, then a self-dual code meetng the bound must be doubly-even (see []). Moeove, fo any nonzeo weght w n such a code, the codewods of weght w fom a 5-desgn [3]. Ths s one eason why extemal codes of length 4m ae of patcula nteest. Unfotunately, only fo m = 1 and m = such codes ae known, namely the [4,1,8] extended Golay code and the [48,4,1] extended quadatc esdue code. Thus the exstence of no othe extemal code of length 4m s known. Fo n = 96, only the pmes, 3 and 5 may dvde the ode of the automophsm goup of the extemal code. We focus ou attenton on the case of an automophsm of ode 15. In [1] t s poved that a bnay doubly-even [96, 48, 0] self-dual code wth an automophsm of ode 15 does not exst. The queston of fndng a doubly-even self-dual [96, 48,16] code fst - 100 -
Факултет по математика и информатика, том ХVІ С, 014 appeas n [4] whee also the fst such code was constucted. In ecent yeas such codes wth an automophsm of ode 3 ae constucted n [5] and [6]; fou codes ae known fom [7]; ten moe codes ae constucted n [8] and a code wth an automophsm of ode 6 35 31s constucted n [9]. We wll constuct many new doublyeven self-dual [96,48,16] codes wth an automophsm of ode 15 usng a method fo constuctng bnay self-dual codes nvaant unde the acton of a cyclc goup of ode pq fo odd pmes p q. The stuctue of the note s as follows. We begn wth shot descpton of the method n Secton (fo moe detals and poves we efe the eade to [1]). In Secton 3 we apply ths method to obtan codes wth n= 96, k = 48 and mnmum dstance 16 havng an automophsm of type 15-(6, 0,, 0).. Self-dual codes wth an automophsm of ode pq fo p < q odd pmes We consde the case = pq fo dffeent odd pmes p and q such that s a pmtve oot modulo p and modulo q. The gound feld s F. Then p 1 q 1 x 1 ( x 1) Q( xq ) ( xq ) ( x) (1 x)(1 x x = = + + + + )(1 + x+ + x ) Q( x), p q whee Q ( x ) s the -th cyclotomc polynomal. Moeove, both Q ( ) p x and Qq ( x ) ae educble ove F snce s a pmtve oot modulo p and modulo q as well. Fnally, f Q( x) = g3( x) gs( x) h1( x) h1( x) ht( x) ht ( x) s the factozaton of the -th cyclotomc polynomal nto educble factos ove F, then these factos have the same degee, φ() ( p 1)( q 1) namely =, whee φ s Eule's ph functon. s + t s + t Let the code C be a bnay self-dual codes possessng an automophsm of ode pq. - 101 -
Факултет по математика и информатика, том ХVІ С, 014 = Ω ΩΩ Ω Ω Ω Ω Ω σ 1 c c+ 1 c+ t 1 1, q c+ tq+ c+ tq+ tp c+ tq+ tp+ c+ tq+ tp+ f whee Ω = (( 1) + 1,, ) ae the pq cycles fo = 1,, c, Ω c+ = ( c + ( 1) q + 1,, c + q) cycles of length q fo = 1,, t q, Ω c+ t ( ( 1) 1,, ) q + = c + tqq + p + c + tqq + p cycles of length p fo = 1,, t p, and Ω c+ t ( ) q+ tp+ = c+ tq+ tp+ ae the fxed ponts fo = 1,, f. Let Fσ ( C) = { v C: vσ = v} and Eσ ( C) = { v C: wt( v Ω) 0 ( mod ), = 1,, c+ tq + tp}, whee v Ω s the estcton of v on Ω. Wth ths notaton we have the followng. Theoem 1 ([1]) The code C s a dect sum of the subcodes Fσ ( C) and Eσ ( C). n Let F be the n -dmensonal vecto space ove the bnay c+ tq+ tp+ f feld F, and let π : Fσ ( C) F be the poecton map,.e., ( π ( v)) = v fo some Ω and = 1,,, c+ tq + tp + f. Clealy, v Fσ ( C) ff v C and v s constant on each cycle of σ. Theoem ([1]) If C s a bnay self-dual code wth an automophsm σ of odd ode then Cπ = π ( Fσ( C)) s a bnay selfdual code of length c+ tq + tp + f. Consde the facto ng [ ]/ 1 R = F x x, whee x 1 s the pncpal deal n F [ x] geneated by x 1. Let x 1 = f ( x) f ( x) f ( x) be the factozaton of x 1 nto 0 1 s - 10 -
Факултет по математика и информатика, том ХVІ С, 014 educble factos f ( x ) ove F whee f ( x) = x 1. Let 0 x 1 x 1 I = be the deal of R geneated by fo f ( x) f ( x) = 0,1,, s. By e ( x ) we denote the geneato dempotent of I ;.e., e ( x ) s the dentty of the two-sded deal I. Wth these notatons we have the followng esult (see [10]). Theoem 3 () R = I0 I1 Is. () I s a feld whch s somophc to the feld F deg ( f ( x )) fo l = 0,1,, s. () e ( x) e ( x ) = 0 fo. (v) s e ( x) = 1. = 0 Thee s a decomposton (see [11]) x 1 = g0( x) g1( x) gm( x) h1( x) h1( x) ht( x) ht ( x), whee s = m+ t and { g0, g1, gm, h1, h1,, ht, ht} = { f0, f1,, fs}. Futhemoe, h ( x ) s the ecpocal polynomal of h ( x ), h h fo = 1,, t and g ( x ) concdes wth ts ecpocal polynomal whee g0( x) = f0( x) = x 1. We denote the feld fo = 0,1,, m, H fo = 1,, t. x 1 h ( x) by x 1 g ( x) by x 1 H fo = 1,, t, and h ( x) G by - 103 -
Факултет по математика и информатика, том ХVІ С, 014 σ ( C) be the shotened code of E ( C) Let E σ obtaned by emovng the last tq q + t p p+ f coodnates fom the codewods c c havng 0's thee. Next we defne a map ϕ : F R by c φ( v) = ( v ( x), v ( x),, v ( x)) R, whee 0 1 c 1-104 - 1 v ( x) = v x and ( v0,, v, c 1) = v Ω. Clealy, ϕ ( C) s a lnea code ove the ng R of length c. Moeove, we have φ( C) = φ( C ) whee the dual code C ove F s taken unde the Eucldean nne poduct, and the dual code φ( C) c n R s taken wth espect to the Hemtan nne c 1 c 1 1 poduct: uv, = uv R, v = v( x ) = v( x ). In patcula, = 0 the code C s self-dual f and only f φ ( C) s self-dual ove R wth espect to the Hemtan nne poduct. Let Cϕ = ϕ( Eσ ( C) ). Snce Eσ ( C) s a bnay quas-cyclc code of length c and ndex c, C ϕ s a lnea code ove the ng R of m t length c. Moeove Cϕ = ( = 0M) ( 1( M = M )), M s a lnea code ove the feld G, = 1,, m, M s a lnea code ove H and M s a lnea code ove H, = 1,, t. Fo the dmensons we have dm E ( C) = dmc = σ φ = 0 whee s t ( p 1)( q 1) = ( p 1) dm M1+ ( q 1) dm M + ( dm M + (dm M + dm M )). s + t = = 3 1 Snce Eσ ( C) s a self-othogonal code, C ϕ s also selfothogonal ove the ng R wth espect to the Hemtan nne poduct. Ths means that M ae self-othogonal codes of length c
Факултет по математика и информатика, том ХVІ С, 014 ove G fo = 1,, m (wth espect to the Hemtan nne poduct) and, fo 1 t, we have M ( M ) wth espect to the Eucldean nne poduct. Ths foces dm M c / fo = 1,,, s and dm M + dm M c. It follows that c c ( p 1)( q 1) c c( pq 1) dm Eσ ( C) ( p 1) + ( q 1) + (( s ) + tc) =. s + t 3. Doubly-even self-dual [96, 48, 16] codes wth an automophsm of type 15 (6, 0,, 0) Let C be a doubly-even self-dual [96, 48, 16] code havng an automophsm of type 15 (6, 0,, 0). The weght dstbuton of such a code has the fom (see [7]) 16 0 4 W( y) = 1 + ( 8086 + α) y + (366643 16 α) y + (366474560 + 10 α) y +, whee α s an ntege paamete. Codes wth α = 36918, 3733, 37608, 37884, 380, 38160, 3898, 38436, 38574, 3871, 38850, 38988, 3916, 3964, 3940, 39540, 39678, 39816, 39954, 4009, 4030, 40368, 40506, 4090, 41334 ae known fom [5]; α = 37500, 3754, 37584, 37598 ae fom [7]; the code n [4] has the weght enumeato coespondng to α = 377. Also the value α = 41106 s known fom [8] and α = 36864, 36876, 36888, 36900, 3691, 36936, 36948, 36960, 3697, 36984 ae fom [9]. We have that M 1 s a Hemtan self-othogonal [6,, ] code ove the feld G1 F 4, M s a Hemtan self-dual [6,3, d ] code ove G F 16, M s a lnea [6, k', d '] code ove H F 16 and M = ( M ) s ts dual code wth espect to the Eucldean nne poduct. Moeove, the code C has a geneato matx n the fom - 105 -
Факултет по математика и информатика, том ХVІ С, 014 1 π ( C ) π 1 ϕ ( M ) 0 1 ϕ ( M ) 0 (1) G =, 1 ϕ ( M ) 0 1 ϕ ( M1) 0 1 1 ϕ ( D) ϕ ( I) genm 1 whee the matx geneates the dual code of M 1 ove G 1, D and I s the dentty matx ove the quatenay feld P 3. Fo the geneato matces of the codes M, M, M and C π we efe the eade to [1]. In shot 47 doubly-even self-othogonal [96,40,0] codes C96,40,1,, C96,40,47 wee constucted and we contnue to add the last 8 ows n G (comng fomϕ 1 1 ( M1) and ϕ ( D1 )) to obtan [96, 48,16] codes. Snce n evey pevous step we mpose the estcton that the mnmum dstance of the code s 0 we cannot gve full classfcaton. We have fou possble geneato matces fo M 1 : e1 0 e1 0 0 0 e1 0 e1 e1 e1 0 G1 =, G =, 0 e1 0 e1 0 0 0 e1 e1 xe1 x e1 0 e1 0 e1 0 0 0 e1 0 0 e1 e1 e1 G3 =, G4 =. 0 e1 0 e1 e1 e1 0 e1 e1 0 e1 e1 Table 1: The values of α and the numbe of [96, 48, 16] codes wth that α obtaned α # α # α # α # α # α # α # 35316 1 36006 46 36486 596 36966 1660 37446 1039 3796 164 38406 16 35376 3601 61 3649 589 3697 1757 3745 1078 3793 179 3841 1 3544 1 36036 53 36516 668 36996 1664 37476 977 37956 167 38436 5 3550 1 3604 66 365 79 3700 1593 3748 945 3796 171 3844 5 3556 36066 74 36546 766 3706 1673 37506 950 37986 14 38466 5-106 -
Факултет по математика и информатика, том ХVІ С, 014 35586 3607 66 3655 738 3703 1597 3751 1036 3799 14 3847 9 3559 4 36096 71 36576 795 37056 1797 37536 870 38016 17 38496 5 356 3 3610 68 3658 798 3706 1819 3754 890 380 114 3850 35646 6 3616 14 36606 918 37086 1686 37566 807 38046 9 3856 4 3565 1 3613 106 3661 936 3709 1613 3757 753 3805 110 3853 5 35676 9 36156 133 36636 95 37116 165 37596 70 38076 79 38556 3 3568 8 3616 134 3664 960 371 1647 3760 746 3808 89 3856 35706 6 36186 145 36666 107 37146 1719 3766 636 38106 74 38586 6 3571 3619 14 3667 1037 3715 1730 3763 696 3811 60 3859 35736 7 3616 197 36696 1174 37176 160 37656 63 38136 63 38616 3574 7 36 19 3670 117 3718 1640 3766 580 3814 57 386 1 35766 13 3646 46 3676 1146 3706 1618 37686 544 38166 38 38646 3577 11 365 31 3673 13 371 1603 3769 569 3817 54 3865 35796 10 3676 38 36756 13 3736 1610 37716 461 38196 5 38706 1 3580 5 368 33 3676 1305 374 1643 377 459 380 6 3871 1 3586 16 36306 95 36786 1410 3766 1345 37746 390 386 33 38736 3583 1 3631 85 3679 1381 377 1467 3775 431 383 37 3874 1 35856 18 36336 376 36816 1360 3796 1359 37776 380 3856 7 3880 1 3586 1 3634 336 368 1400 3730 1440 3778 40 386 9 3886 1 35886 14 36366 356 36846 1446 3736 1401 37806 96 3886 6 38886 1 3589 0 3637 381 3685 1435 3733 1476 3781 310 389 3 38916 1 35916 6 36396 378 36876 159 37356 156 37836 63 38316 4 389 1 359 4 3640 414 3688 1553 3736 136 3784 77 383 35946 3 3646 510 36906 1467 37386 1164 37866 53 38346 0 3595 4 3643 485 3691 155 3739 13 3787 9 3835 1 35976 46 36456 59 36936 158 37416 100 37896 9 38376 10 3598 37 3646 579 3694 1574 374 111 3790 40 3838 16 Afte consdeng all pemutaton τ S6 of the columns of G1,, G4 and all ght cyclc shft n all 6 sx columns we found codes only when gen M1 = G3. The next theoem s a summay of the esults we have obtaned. Theoem 4 Thee exst at least 114966 bnay doubly-even [96, 48,16] self-dual codes wth an automophsm of type 15-(6, 0,, 0). 114171 of the obtaned codes have automophsm goups of ode 15; 763 have automophsm goups of ode 30, and 3 codes have goups of ode 45. The 19 values of the paamete α n the weght dstbuton W( y ) and the numbe of the constucted codes ae dsplayed n Table 1. Of all 19 only 8 values: 36876, 3691, 36936, 3697, 3733, 380, 38436, 3871 wee pevously know, so we obtan 11 new values of the paamete. - 107 -
Факултет по математика и информатика, том ХVІ С, 014 REFERENCES 1. Bouyukleva St., W. Wllems and N. Yankov, On the Automophsms of Ode 15 fo a Bnay Self-Dual [96, 48, 0] code. // axv:1403.4735 [cs.it], 014.. Rans E.M., Shadow bounds fo self-dual-codes. // IEEE Tans. Infom. Theoy, vol. 44, pp. 134 139, 1998. 3. Assmus E.F. and H.F. Mattson, New 5-desgns. // J. Combn. Theoy, vol. 6, pp. 1 151, 1969. 4. Fet W., A self-dual even (96, 48, 16) code. // IEEE Tans. Infom. Theoy, vol. 0, pp. 136 138, 1974. 5. Dontcheva R., Doubly-even self-dual code of length 96. // IEEE Tans. Inf. Theoy, vol. 48, no., pp. 557 561, 00. 6. Yogova R. and A. Wassemann, Bnay self-dual codes wth automophsms of ode 3. // Des. Codes Cyptog., vol. 48, no., pp. 155 164, 007. 7. S. T. Doughety, T. A. Gullve, and M. Haada, Extemal bnay self-dual codes. // IEEE Tans. Inf. Theoy, vol. 43, pp. 036 047, 1997. 8. Kaya A. and B. Yldz, Extenson theoems fo self-dual codes ove ngs and new bnay self-dual codes. // axv:1404.0195 [cs.it], 014. 9. Kaya A., B. Yldz, and I. Sap, New extemal bnay self-dual codes of length 68 fom quadatc esdue codes ove F + u F. + u F // Fnte Felds The Appl., vol. 9, pp. 160 177, 014. 10. Huffman W.C., V. Pless, Fundamentals of Eo-Coectng Codes, Cambdge Unvesty Pess, Cambdge 003. 11. Lng S., P. Sole, On the algebac stuctue of quas-cyclc codes I: Fnte felds. // IEEE Tans. Infom. Theoy, vol. 47, pp. 751 760, 001. - 108 -