QT codes. Some good (optimal or suboptimal) linear codes over F. are obtained from these types of one generator (1 u)-

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1 Mathematca Computato March 03, Voume, Issue, PP-5 Oe Geerator ( u) -Quas-Twsted Codes over F uf Ja Gao #, Qog Kog Cher Isttute of Mathematcs, Naka Uversty, Ta, 30007, Cha Schoo of Scece, Shadog Uversty of Techoogy, Shadog Zbo, 5509, Cha #Ema: agao@maakaeduc Abstract Ths documet gves the mmum geeratg sets of three types of oe geerator ( u)- quas-twsted (QT) codes over F uf, u 0 Moreover, t dscusses the geeratg sets ad the ower bouds o the mmum ee dstace of a speca cass of type oe geerator ( u)- QT codes Some good (optma or suboptma) ear codes over F are obtaed from these types of oe geerator ( u)- QT codes Keywords: Oe Geerator ( u)- Quas-Twsted Codes; Mmum Geeratg Sets; ear Codes A INTRODUCTION Recety, t has bee show that codes over fte rgs are a very mportat cass ad may types of codes wth good parameters ca be costructed over rgs [, 3, 6, 9] atey, there are some research o codg theory over fte cha rg Fq ufq u Fq, where q s a postve power of some prme umber p ad s, the oy fte cha rg wth character p [-,6-8] I [8], Sh vestgated the structura propertes ad the mmum geeratg sets of costacycc codes over Fq ufq u Fq Abuarub gave the cassfcato of ( u)- cosacycc codes wth arbtrary egth over F uf[] I [7], Ka studed the structura propertes of ( u)- cosacycc codes over F uf u F q q q Quas-cycc (QC) codes over commutatve rgs costtute a remarkabe geerazato of cycc codes More recety, they produce may codes over fte feds whch meet the best vaue of mmum dstaces of the same egth ad dmeso [3, 6, 9] Quas-twsted (QT) codes as a geerazato of QC codes, have some good agebrac structures as we produce may good codes over fte feds [4, 5] I ths paper, we may research oe geerator ( u)- QT codes over F uf The rest of the preset paper s orgazed as foows I Sec, we survey some we-kow resuts reated to our work I Sec3, we gve the geeratg sets of three types of ( u)- QT codes over F uf I Sec4, we cosder a speca cass of QT codes, whch ca ead to costructo of some good codes over F PREIMINARIES et R F uf, 0 R ( c0, c,, c ) c R, 0,, A oempty set C of R s caed a ear code f ad oy f C s a R -submodue of R Defe a ear operator T o R T( c) (( u) c, c0,, c ) for each c ( c0, c,, c ) R A ear code C s caed a ( u)- costacycc code of egth over R f ad oy f T C u ad et S R / ( x ( u)) Defe a R -modue somorphsm as foows : R S ( c, c,, c ) c c x c x 0 0 Oe ca verfy that C s a ( u)- costacycc code f ad oy f s a dea of S I ths paper, we equa - -

2 ( u)- costacycc code to the dea of S The foowg two emmas w be used the dscussg mmum geeratg sets of oe geerator ( u)- QT codes emma (cf [] emma) et C be a ( u)- costacycc code of egth over R The C ( g( x) up( x), ua( x)), where a( x) g( x) ( x ) over F ad deg a( x) deg p( x) emma (cf [] Coroary ) et C ( g( x) up( x), ua( x)) be a ( u)- costacycc code egth over R, where e mad gcd(, m) The C has the foowg three types: () C ( g( x)), where g( x) ( x ) over F ; () C ( ug( x)), where g( x) ( x ) over F ; (3) r e e C f f f r, where f ( x) ( x ) over F ad there exsts a, We ca the above three types ( u)- costacycc codes A type, A type ad B type, respectvey Defe the ee weght W of the eemets 0,,u, u as 0,,,, respectvey Moreover, the ee weght of a - tupe R s the sum of the ee weghts of ts compoets The Gray map seds the eemets 0,, u, u of R to (0,0), (0,), (,), (,0) over F, respectvey It s easy to verfy that s a ear sometry from R (ee dstace) to F (Hammg dstace) 3 ONE GENERATOR ( u)-qt CODES A ear code C over R of egth N s caed quas-twsted (QT) code f t s varat udert for some postve teger The smaest T C s caed the dex of C Ceary, s a dvsor of N et N Defe a oe-to-oe correspodece N : R S ( a, a,, a ; a, a,, a ;; a, a,, a ) 0,0 0, 0,,0,,,0,, f ( x) ( f ( x), f ( x),, f ( x)) 0 where f ( x ) a x 0, for 0,,, The C s equvaet to sayg that, for ay f ( x) ( f0( x), f( x),, f ( )) ( ) x C, x f ( x) Therefore, C s a QT code f ad oy f s a Rx-submodue [ ] of S et C S f ( x), where f( x) s defed as above The C s caed a oe geerator ( u) -QT code For smpcty, we deote C ( f0( x), f( x),, f ( x)) et C be a oe geerator ( u) -QT code of egth wth dex over R, where e mad gcd(, m) Assume that C ( f0( x), f( x),, f ( x)), where f ( x) S, 0,,, for each 0,,,, defe a Rx [ ] -modue homomorphsm as foows : S S ( f ( x), f ( x),, f ( x)) f ( x) 0 The s a dea S, e, s a ( u) -costacycc code of egth over R If s the type of A, A or B, the we ca C s the A, A or B type oe geerator ( u) -QT code, respectvey Theorem 3 et C be a A type oe geerator ( u) -QT code of egth wth dex over R geerated by G ( g0, g,, g ), where e m ad gcd(, m), [ ] g F x, 0,, et g gcd( g0, g,, g, x ), h ( x ) g, degh r, f g g, F uf0, uf,, uf The the mmum geeratg set of C s S S, where r S G, xg,, x r G S F, xf,, x F ad Proof et cx ( ) be a code word of C The there exsts f ( x) S Usg Eucdea dvso, there are poyomas Q, R R[ x] c( x) f ( x) G f ( x)( g0, g,, g ) () f () x Q h R ()

3 where deg Q r, R 0 or deg R r Therefore from () ad (), we have c( x) Q h( g, g,, g ) R ( g, g,, g ) 0 0 ceary, R ( g0, g,, g ) Spa( S) Sce ( g0, g,, g ) g( f0, f,, f ), we have Q h( g, g,, g ) Q hg( f, f,, f ) (3) 0 0 Note that u x gh S Therefore (3) Q ( uf0, uf,, uf ), e, Q h( g, g,, g ) Spa( S ) 0 That mpes that S S geerates C Next, we w prove ( ) ( ) 0 et e x e0 x e x e x Spa S Spa S Spa S Spa S ( ) ( ( ), ( ),, ( )) ( ) ( ) Sce e( x) Spa( S), t foows that O the other had, sce e( x) Spa( S), e x g x x (4) r ( ) ( 0 r ), 0,,, e x uf x x (5) r ( ) ( 0 r ) From (5), for each 0,,,, ue ( x) 0 whch mpes that 0 or 0, 0,,, r Assume that M () x x x r 0 r M () x x x r 0 r The g M( x) uf M( x) Thus ug ( M( x) hm ( x)) u f M( x) 0 Usg the facts e ( x) Spa( S) ad M( x) F[ x], we have that M( x) hm ( x) 0 So for each 0,,, r, ad 0,,, r, 0 It meas that Spa( S ) Spa( S ) 0 Theorem 3 et C be a A type oe geerator ( u) -QT code of egth wth dex over R geerated by G ( ug0, ug,, ug ), where e m ad gcd(, m), [ ] g F x et g gcd( g0, g,, g, x ), ( r h x ) g, degh r S G, xg,, x G The the mmum geeratg set of C s Proof et cx ( ) be a codeword of C The there exsts f ( x) S Usg Eucdea dvso, there are poyomas Q, R R[ x] c( x) f ( x) G f ( x)( ug0, ug,, ug ) (6) f () x Q h R (7) Where R 0 or deg R r Therefore from (6) ad (7), we have c( x) ( uq hg, uq hg,, uq hg ) R ( ug, ug,, ug ) 0 0 Note that u x gh ad gg S, 0,,, Therefore c( x) R ( ug0, ug,, ug ) Whch mpes that S geerates C By the costructo of S, S s mmum Thus S s the mmum geeratg set of C Theorem 33 et C be a B type oe geerator ( u) -QT code of egth wth dex over R geerated by G ( q0 fg, q fg,, q fg), where e m ad gcd(, m), q [ ] F x, 0,,, ad g F [ x] wth the maxma degree satsfyg f g ( x ) et h ( x ) g, deg g r, deg f t, gcd( q, h) ad r F uq 0 f, uq f,, uq f The the mmum geeratg set of C s S S, where S G, xg,, x G ad t S F, xf,, x F Proof We ust prove Spa( S ) Spa( S ) 0 et e x e0 x e x e x Spa S Spa S ( ) ( ( ), ( ),, ( )) ( ) ( ) Sce e( x) Spa( S), t foows that

4 O the other had, sce e( x) Spa( S), e x q fg x x, (8) r ( ) ( 0 r ) 0,,, e x uq f x x, (9) t ( ) ( 0 t ) 0,,, From (9), for each 0,,,, ue ( x) 0 whch mpes that 0 or u, 0,,, r Assume that M () x x x r 0 r M () x x x t 0 t The q fgm( x) uq fm ( x) Thus uq fg( M( x) hm ( x)) u f M( x) 0, whch mpes that q ( fg x ) or M( x) hm ( x) 0 Sce gcd( q, h), t foows that q fg does ot dvde x Therefore M( x) hm ( x) 0 So for each 0,,, r ad 0,,, r t, we have 0 Spa( S ) Spa( S ) e, I the rest of ths secto, we preset some exampes to ustrate the appcatos to these theorems Exampe 34 Takg 3, wth g0 x ad g x, we get a A type oe geerator ( u) -QT code C of egth 3 6 wth dex over R Sce g x, f0 ad f x, from Theorem 3 the geeratg set of C s ( g0, g), x( g0, g) uf 0, uf, whch mpes that C 4 By the Gray map, we get s a optma [,5,4] ear code over F Takg 3, 3 wth g0 x x x, g x x x ad g x, f0 x x, f x x ad f,we ca get a A type oe geerator ( u) -QT code C of egth 9 wth dex 3 over R Sce g x, from Theorem 3 the geeratg set of s ug 0, ug, ug, x ug0, ug, ug, whch mpes that C over R By the Gray map, s a optma [8,,] ear code over F 6 3 Takg 9, wth f x, g ( x )( x x ), q0 x ad q x x, we get a B type oe geerator ( u) -QT code C of egth 9 8 wth dex over R From Theorem 33, we have the geeratg 5 set of C ( q0 fg, q fg), x( q0 fg, q fg, q fg) uq0 f, uq f, uq f, x uq0 f, uq f,, x uq0 f, uq f, whch mpes 6 that C 4 By the Gray map, we get s a [36,0,8] ear code over F 4 A SPECIA A TYPE I ths secto, we dscuss a speca cass of A type oe geerator ( u) -QT codes We determe the mmum geeratg sets amog them ad troduce a ower boud for the mmum ee dstace Theorem 4 et C be a A type oe geerator ( u) -QT code of egth wth dex over R geerated by G ( ugf 0, ugf,, ugf ), where e m ad gcd(, m), g, f F [ x], g ( x ) ad gcd( f,( x ) g), r 0,, etdeg g r The the mmum geeratg set of C s S G, xg,, x G Moreover, the mmum ee dstace of C s d d, where C ( ug) O the other had, f for each 0,,,, gcd( f,( x ) g), the ( ug) ( ugf ) It foows that C for each 0,,, et c( x) ( ugf0a( x), ugfa( x),, ugf a( x)) be a codeword of C The ugfa( x) 0 f ad oy f ( x ) gfa( x) f ad oy f gh gf a( x) f ad oy f h f a( x )) f ad oy f h a( x ) It meas that f ugf a( x) 0, the cx ( ) 0, 0,,, Therefore for ay ozero codeword cx, ( ) ( cx ( )) 0 mpes that d d r Proof from Theorem 3, oe ca verfy that the mmum geeratg set of C s S G, xg,, x G Usg Theorem 4, some exampes of ths famy that yed good (optma or suboptma) ear codes are gve Exampe Takg 3, wth g x, f0 x x ad f x x, we get a A type oe geerator ( u) -QT 3- code C of egth 3 6 wth dex over R From Theorem 4, C 4ad d 4 8 Sce there are some codewords wth ee weght 8, the mmum ee weght of C s 8 actuay, e, C s a (6,4,8) ear code over R By the Gray map, we get a[,,8] ear code, whch s optma over F

5 4 Takg 4, wth g x, f0 x x ad f x x, by the hep of Theorem 4 ad the Gray map, we have a suboptma [6,3,8] ear code over F 4 Takg 4, 3 wth g x, f0 x x, f x x ad f x, by the hep of Theorem 4 ad the Gray map, we have whch s a optma [4,3,] ear code over F ACKNOWEDGMENT The authors are deepy debted to the referees ad wsh to thak for ther mportat suggestos ad commets Ths research s supported part by the Natura Scece Foudato of Shadog provce (Grat No ZR0AQ004) REFERENCES [] Abuarub, T, Sap, I Cycc codes over the rg Z uz ad Z uz u Z Desgs, Codes ad Cryptography 4(007): [] Abuarub, T, Sap, I Costacycc codes over F uf Joura of the Frak Isttute 346(009): [3] Ayd, N, Ray-Chaudhur, D K Quas-Cycc Codes over Z4 ad Some New Bary Codes IEEE Tras Iform Theory 48(00): [4] Ayd, N, Sap, I, Ray-Chaudhur, D K The structure of -geerator quas-twsted codes ad ew ear codes Desgs, Codes ad Cryptography 4(00): [5] Daskaov, R N, Guver, T A New quas-twsted quaterary ear codes IEEE Tras Iform Theory 46(000): [6] Guver, T A, Harada, M Codes over F3 uf3 ad mprovemets to the bouds o terary ear codes Desgs, Codes ad Cryptography (00): m [7] Ka, X, Zhu, S,, P ( u) -costacycc codes over F [ u] / ( u ) Joura of the Frak Isttute 347(00): p [8] Sh, M, Zhu, S Costacycc codes over rg F uf u F Joura of Uversty of Scece ad Techoogy of Cha39(009): q q q [9] Sap, I, Abuarub, T, Ydz, B Oe geerator quas-cycc codes over F uf Joura of the Frak Isttute 349(0): 84-9 Authors Ja GAO was bor Shadog, Cha, 988 He receved hs BS ad MS degree Apped Mathematcs from Scece Schoo of Shadog Uversty of Techoogy Currety he s a doctora studet Cher Isttute of Mathematcs Hs research terests are codg theory ad formato theory Qog KONG was bor Shax, Cha, 988 She graduated wth a MS degree Apped Mathematcs from Chagzh Coege Shax, ad currety she s a master studet Scece Schoo of Shadog Uversty of Techoogy Her research terest s codg theory