SOME NEW SELF-DUAL [96, 48, 16] CODES WITH AN AUTOMORPHISM OF ORDER 15. KEYWORDS: automorphisms, construction, self-dual codes

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1 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 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/

2 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

3 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 s 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

4 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

5 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

6 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 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

7 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]) W( y) = 1 + ( α) y + ( α) y + ( 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, ae known fom [5]; α = 37500, 3754, 37584, ae fom [7]; the code n [4] has the weght enumeato coespondng to α = 377. Also the value α = s known fom [8] and α = 36864, 36876, 36888, 36900, 3691, 36936, 36948, 36960, 3697, 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

8 Факултет по математика и информатика, том ХVІ С, π ( C ) π 1 ϕ ( M ) 0 1 ϕ ( M ) 0 (1) G =, 1 ϕ ( M ) 0 1 ϕ ( M1) 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 e e1 0 e1 e1 e1 0 G1 =, G =, 0 e1 0 e e1 e1 xe1 x e1 0 e1 0 e 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 #

9 Факултет по математика и информатика, том ХVІ С, 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 bnay doubly-even [96, 48,16] self-dual codes wth an automophsm of type 15-(6, 0,, 0) 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

10 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: [cs.it], Rans E.M., Shadow bounds fo self-dual-codes. // IEEE Tans. Infom. Theoy, vol. 44, pp , Assmus E.F. and H.F. Mattson, New 5-desgns. // J. Combn. Theoy, vol. 6, pp , Fet W., A self-dual even (96, 48, 16) code. // IEEE Tans. Infom. Theoy, vol. 0, pp , Dontcheva R., Doubly-even self-dual code of length 96. // IEEE Tans. Inf. Theoy, vol. 48, no., pp , Yogova R. and A. Wassemann, Bnay self-dual codes wth automophsms of ode 3. // Des. Codes Cyptog., vol. 48, no., pp , S. T. Doughety, T. A. Gullve, and M. Haada, Extemal bnay self-dual codes. // IEEE Tans. Inf. Theoy, vol. 43, pp , Kaya A. and B. Yldz, Extenson theoems fo self-dual codes ove ngs and new bnay self-dual codes. // axv: [cs.it], 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 , Huffman W.C., V. Pless, Fundamentals of Eo-Coectng Codes, Cambdge Unvesty Pess, Cambdge Lng S., P. Sole, On the algebac stuctue of quas-cyclc codes I: Fnte felds. // IEEE Tans. Infom. Theoy, vol. 47, pp ,