March 23, TiCC TR Generalized Residue Codes. Bulgarian Academy of Sciences, Bulgaria and. TiCC, Tilburg University

Transcription

1 Tburg cere for Creave Copug P.O. Bo 90 Tburg Uversy 000 LE Tburg, The Neherads hp:// Ea: Copyrgh S.M. Doduekov, A. Bojov ad A.J. va Zae 00. March, 00 TCC TR Geerazed Resdue Codes Bugara Acadey of Sceces, Bugara ad TCC, Tburg Uversy S.M. Doduekov, A. Bojov ad A.J. va Zae

2 Geerazed Resdue Codes S.M. Doduekov, A. Bojov ad A.J. va Zae

3 Absrac A geera ype of code C,, s roduced as a sraghforward geerazao, q, of he we-kow quadrac resdue codes, usg he foras of geerag poyoas over fe feds. Apar fro he paraeer whch has o be a dvsor of, where s he Euer fuco, hese codes are defed over a arbrary fed GF( q) ad have a arbrary egh, wh (, q). I hs sese hey ca be see as a geerazao of quadrac resdue codes, geerazed quadrac resdue codes, Q-codes, resdue codes ad of duadc, radc ad poyadc codes.

4 Coes Iroduco. Preares ad he defo of geerazed resdue codes. The group U ad s subgroups of -resdues. Soe spe properes of geerazed resdue codes. Mu dsaces geerazed resdue codes. The speca case = p 6. The speca case = p 7. The speca case = p 8. Eeded geerazed resdue codes ad he case = Refereces

5 Iroduco Lear cycc codes beog o he os wdey suded agebrac codes he eraure. I hs Repor, q s ea o be he power of soe pre, hough very ofe we sha dea wh q as f were a sge pre. A code C of egh over he fed GF(q), wh (, q), s caed ear f s a ear space over GF(q), ad s caed cycc f s vara uder cycc peruaos. Ths as codo eas ha f c ( c0, c,..., c ) C, he c = ( c, c0,..., c ) C. Here, he copoes c, 0, are a fro GF (q). The soorphs whch aps c oo c 0 c... c he poyoa rg R : GF ( q)[ ] /( ) yeds a cycc prcpa dea ha rg, geeraed by a uque oc poyoa g() of owes degree, wh g(). Usuay, he code C ad hs dea are defed. Very ofe a cycc code s defed by prescrbg he zeros of s geeraor poyoa g() (he zeros of he code) soe eeso fed of GF (q). If g h( ), he h() defes he frs row of a cycc pary check ar of C, ad s herefore caed he check poyoa. A poyoa c() correspods o a codeword of C, f ad oy f c h( ) 0 R. The poyoa h() ca aso be erpreed as he geeraor poyoa of a code whch s equvae o he dua code C. Quadrac resdue codes (QR-codes) are oray roduced as foows. Take for a pre p ad ake for q aso a pre dffere fro p. Moreover, oe requres ha q s a quadrac resdue od p. Le Q be he se of ozero squares GF( p) ad N he se of osquares. Le be a prve -h roo of uy soe eeso fed of GF (q). Oe defes he poyoas g : ( ) Q ad g : ( ). N These poyoas have coeffces ( ) g _ GF (q), ad hey sasfy g. Oe e defes cycc codes Q, Q, N ad N geeraed by he poyoas, ( ) ( ) g, g ad ( ) g, respecvey. A four codes are caed quadrac resdue codes over GF (q) of pre egh ( p). As aready rearked, hese codes are suded esvey, e.g. wh respec o her u dsace, auoorphs Ths repor s based o a paper of he frs auhor: S.M. Doduekov, Resdua codes, Psca Suda Maheaca Bugarca, (98), pp. ( Russa). _ g

6 group, geerag depoe, dua code, eeded code, ec. Quadrac codes have rae cose o ½, ad ed o have hgh u dsace a eas for o oo arge vaues of. As for pracca reasos, quadrac codes are easy o ecode ke a cycc codes, bu geera dffcu o decode. For a hese properes ad reaed aspecs, we refer o [, 9,, ] ad o he eraure eoed hese books. Va L ad MacWas [9], Cao [], Desare [] ad Ward [6] geeraze he above cocep of quadrac codes for code egh p ad for arbrary feds GF( q ), ( q, p). I urs ou ha he ehods ha are used o hade quadrac resdue codes ca easy be geerazed o gve a aaogous heory for hese so-caed geerazed quadrac resdue codes (GQRcodes). Berekap [, Seco.] defes e - resdue codes, whch for e are deca o quadrac resdue codes, ad shows ha he u dsace of such codes sasfes he equay kow as he square roo boud he case of quadrac resdue codes. Bary QR-codes ( q or q ) are he os suded quadrac resdue codes by far. Aso codes over GF() are cosdered occasoay.these are caed Pess syery codes (cf. []). For q e s kow case of quadrac resdue codes. Pess [] suded Q-codes whch coa as a subcass QR-codes over GF(). A dffere kd of geerazao of QR codes for he duadc codes, roduced by Leo, Masey ad Pess [7], where hese are oy defed for he bary case. Sd [6] furher geerazes hs oo for arbrary fe feds. Furher geerazaos of hs ype are radc codes by Pess ad Rushaa [] ad poyadc codes by Pess ad Bruad []. The duadc codes [7] are defed for egh... p p p, where each p s pre ad cogrue o ± od 8. Le S be he fay of cycooc coses {0} od ad e S ad S be subfaes of S wh S S S, S S. Uder cera codos, he poyoas e, {, }, are he depoe geeraors of duadc j S j codes C ad C. I case ha s equa o soe pre p, he faes S ad S are deca o he ses of quadrac resdues ad o-quadrac resdues od, respecvey, ad oe obas he bary quadrac resdue codes of egh. I [] ad [] he oo of duadc code s furher geerazed for spgs of S o hree ad o subfaes, gvg rse o radc ad -adc codes, respecvey. The codes cosdered [] are of pre power egh. The sae hods for he poyadc codes suded by Shara, Baksh ad Raka [7]. They prese ecessary ad suffce codos for he esece of hese codes. The codes suded [] are -adc codes of pre egh over GF( q). I hs Repor we sha roduce he oo of geerazed resdue codes (GR-codes). Such codes cosue a sraghforward geerazao of quadrac resdue codes, sce her geerag poyoas are of he for g ( ), where K sads for soe * subgroup of U (: Z ), he upcave subgroup of Z, of de, or for oe of s coses. Noce ha for p, Q s he subgroup of quadrac resdues of U, whe N s K p

7 s oy cose. Therefore, by akg K : Q whch has de wh respec o oba a speca GR-code wh. I geera, he vaue of he code egh, he uderyg fed GF ( q) ad he vaue of he paraeer are arbrary, apar fro he codo ha has o be a dvsor of ad q has o be a eee of K. I Seco we prese soe geera defos ad oos. I Seco we prese soe we-kow facs abou he group, he upcave group of egers od U whch are pre o. We aso derve a few ess kow properes of whch w be apped he e secos ad aso a e repor deag wh -resdue codes whch cosue a subcass of he GR-codes, defed by he requree ha K s equa o he subgroup U of -powers of he eees of U, for soe vaue of. We ca hs parcuar subcass he cass of -resdue codes or ore geeray, whe we do o specfy he vaue of, he cass of resdue codes. Berekap [, Seco.] roduced he er e-h resdue code for a ype of code whch s deca o our e- resdue code he case ha s a pre. For hs reaso we chose he ae geerazed resdue codes for he codes cosdered hs Repor. I Seco a uber of spe properes of GR- codes are dscussed. The subjec of Seco s a ower boud for he u dsace of he odd-wegh codewords of a GR- code. Ths ower boud geerazes a we-kow resu for QR-codes ad her geerazaos. Secos, 6 ad 7 dea respecvey wh he speca cases U p, p ad p. These cases share he propery ha he group U s cycc, whch appears o be a facag feaure. I Seco 8 a eape s cosdered whch s o covered by hese hree speca cases,.e.. For hs -vaue he group U s o cycc. The eape for aso serves as a usrao of he oo of eeded GR- code. I a e repor we sha coduc a coser vesgao wh respec o GR-codes based o ocycc groups ad parcuar o cases where K s chose o be equa o. Oher opcs whch w ge our aeo he s he cosruco of depoe geeraors ad he auoorphs groups of GR-codes. U U p U, we. Preares ad he defo of geerazed resdue codes p p Le p p..., where, p,, p are pre ubers, ad e q be a arbrary pre (power), wh q p,. Hece, (, q). Le furherore r := ord (q) be he upcave order of q oduo,.e. r s he eas posve eger sasfyg r q od. (.) I oher words, r s he order of q he upcave group U deoed by (cf. defo (.)). * Z, whch usuay s 6

8 Le () be he - h cycooc poyoa over he fed of raoas. The () s a dvsor of, as foows fro he we kow epresso d / d. I he e, we sha wre ( ) P( ). (.) Moreover, deg () s equa o (), where s he we-kow Euer fuco. Sce (.) hods Z [], aso hods Z q [] ad hece, we ca cosder () as a poyoa over he fe fed GF (q). More specfcay, we ca wre (cf. [8, Theore.7]) deg ( ) r, (.) for soe eger, ad we have he foowg facorzao F[] F F... F (.) o poyoas F ( ),, a of degree r as defed (.), whch are rreducbe over he fed F := GF (q) (cf. aso [] where he case s pre s cosdered). We aso roduce he upcave group of posve egers od, whch are copre wh,.e. he upcave group of he rg Z, represeed by Eape. G : U { a a, ( a, ) }. (.) Take 6 ad q 7. The upcave order of 7 od 6 s equa o r. Le be a prve 6-h roo of soe eeso fed of GF (7). I hs case s o ecessary o eed GF(7), sce ord 7 () = 6, ad so we ca ake 6 GF(7). Hece, we have ( ) ( )( ) = ( )( ) =. A aerave dervao s by appyg ad usg (). We aso kow ha 6() s a dvsor of 6. I parcuar we ca wre 6, wh, ad 6 ) k (, whch devers he epresso for. k 6 Sce deg., foows ha. I appears ha GF( 7)[ ] we ca 6 facorze 6 () as 7

9 6 F F : ( )( ). Eape. Ne we cosder he case, so U {,, 7, }. Le be a prve -h roo of a eeso fed of. The -h cycooc poyoa over equas 7 ( )( )( )( ) = ( )( ). If we cosder () as a poyoa GF( )[ ], we have he facorzao F F : ( )( ), where F () ad F () are rreducbe ad boh of degree. Sce od, we have r ad hece, whch usraes aga he facorzao (.). Sar resus ca be derved for q 7 ad q. I GF( 7)[ ] as we as GF[ ] we have r ad, sce 7 ad od. The facorzaos are respecvey ad ( )( ) ( )( ). Cosder he group G defed (.). The a subgroup H G coag q s he cycc group geeraed by q,.e. H q {, q, q,..., q r }, (.6) where r ord (q). Sce he facorgroup G / H has order G / H =, we ca wre 8

10 G H H H... H (.7) where he coses H are o-ersecg cycooc casses, defed by H H wh represeave eees,,..,. Now cosder he poyoa P ( ), (.8) H wh {,,..., }, ad where s a prve -h roo of uy soe approprae eeso fed of F GF(q). Ceary deg () s equa o H = r ad P. Lea. By approprae deg he poyoas F () (.), oe has P F, for a {,,..., }. Proof We wre he poyoa (.8) as r r P ( ) cr... c c H cr c c0 where he coeffces,...,, are syerc fucos of he eees, H. Sce he eees of he se H are perued aog each oher uder q j upcao by q, foows ha c c. Hece, c j F for a j, 0 j r. So P F[ ], for. Moreover, a for j. Therefore, j P P () are dvsors of ( ) ad P () ad P j () have o coo zeros P s a facorzao of 0, () o poyoas over F of degree r. Eq. (.8) s a sar facorzao o rreducbe poyoas over F of degree r. Sce a decoposo of () o rreducbe poyoas over soe fed fed s uque, foows ha he () are rreducbe ad equa o he poyoas () fro (.8) soe order. F Eape. Take ad q 7. We have U {,, 7, } ad () (cf. Eape.). Sce H H {, 7} ad H {, }, foows ha P ( ) ( )( ) ( P ), 9

11 ( )( ) ( P ), where we used ad =. 6 I order o deere he cosas he above poyoas, we pu :. The s a roo of he equao 0. I GF( 7 )[ ] we have ( )( )( ). Sce, does o correspod o a prve roo, we have eher or. The choce gves P, P. The choce gves a equvae resu, erchagg he dces ad. Coparg hs wh he poyoas F () ad F () (cf. aga Eape.), usraes Lea.. Fay, we prese a eape whch shows ha for a gve vaue of ad fed vaues of r ad, he cycooc poyoa ( ) ca facorze o facors of degree r, bu over dffere feds. Eape. For, we have, U {,,,} ad (). Accordg o eq. (.), he poyoa facorzes o /r rreducbe poyoas of degree r over ay fed GF( q ) wh ( q,) ad ord ( q) r. As a eape we ake successvey respec o. I he case q q ad we ca wre ( ) ( )( ), q whch have boh order wh where s a zero of he rreducbe (over GF()) I he case q 9, we have he facorzao poyoa. ( ) ( )( ), wh a zero of he rreducbe poyoa (over GF() ). If we ake q, we have ord () = ad, ad so ( ) s rreducbe GF [ ]. Soehg sar hods for q whch aso has order wh respec o, fro whch we cocude ha s rreducbe GF [ ]. Suppose s,, ad e K be a subgroup of G of de such ha H K G. (.9) 0

12 Fro he assupos foows ha G = () = r = rs, K = rs, H = r. (.0) Because of (.9) we ca wre, by reabeg he H - coses, K H H... H s (.) ad aso G y K y K... y K = K K... K (.) for soe suabe eees y, y,., y G. Cobg (.) ad (.), ad usg H H, provdes us wh G, j s, j wh : y. Ths s equvae o (.7)., j j H, (.) Sce s a prve -h roo of uy soe eeso fed of F, he poyoas g ( ) = F (), K s k j k, (.) of degree rs, are dvsors of ad have coeffces F, as ca be prove a sar way as Lea.. (cf. []). The poyoas F j (),, () cosue soe subse of he se of poyoas F (),., F () (or of he poyoas P,..., P ) roduced (.) ( (.8)). A spe cosequece s g Fj s. (.) Defo. The cycc code C, q, of egh over F wh geeraor poyoa g s caed a geerazed resdue code for ay {,,..., }. If he group K (.9) s deca o a subgroup U, where s he eas vaue wh hs propery, we sha aeravey speak of a -resdue code.

13 Defo. The eve-ke wegh subcode of, geeraed by ( ) g ( ad deoed by s caed he epurgaed code, whe code. C, q, ) C, q, sef hs coe s caed he augeed C, q,, Rearks Fro he defo of he group K as a subgroup of G wh de, foows ha he codes C are oy defed for. I Defo. we use he er eve-ke,, q, sce oy he bary case he weghs of he codewords are reay eve. I cases whe w o gve rse o cofuso, we sha occasoay o he subdces, q ad. A speca subcass of GR -codes arses whe we choose K equa o a subgroup U cossg of he -powers or he -resdues U for soe -vaue. We sha ca he resug codes -resdue codes. Acuay, hey geeraze sar codes roduced by Berekap [, Seco.] for he case ha s equa o soe pre p. More precsey, for C p, q, he group K s he subgroup of - powers G : U p,.e. he subgroup usuay caed G, of hose eees r for whch he equao r od p has a eas oe souo G. Sce geera - powers are copued by reduco od, we sha fro ow o o dsgush bewee he ers -powers ad -resdues (cf. aso [6, Seco.], where he er -h power resdue s used). We sha prove ow ha case of, he dey K U p hods. C p, q, Theore. If s a pre, he group K as defed (.9) as a group of de wh respec o G, s cycc ad cosss of he -resdues of G. Proof Sce G / K, we have for a coses K,, ha K K K. Hece, g K for a g G ad so G K. Now, suppose U s cycc. So, here ess a eee a wh a. Fro U = rs, we cocude ha he eees a, a rs U,.., a are a dffere eees of G. So, G : K. I s we kow ha a group U wh p pre, s cycc. Ths proves he above saee. p I parcuar, whe akg p ad, we oba he we-kow quadrac resdue codes or QR codes (cf. Seco ). U

14 . The group U ad s subgroups of -resdues I he proof of Theore., we apped he propery ha s cycc. For he sake of coveece ad for fuure appcaos, we sha prese ad prove a few eas coag a uber of properes of he groups U ad her subgroups U of resdues. As for he proofs of hose properes whch are raher we-kow, we sha refer o [6]. I such proofs oe ca epo he hooorphs : G G for a abea group G. I hs Repor however, we sha o do so. U p Lea. Le be odd, ad e A be he se of odd egers U ad B he se of eve egers. The he foowg reaos hod: ()A=B, ad he reaos b a, f a, ad b ( a), f a, defe a oe-o-oe appg fro A o B; () U =U, ad U A C, where C s a se of egers c (, ), defed by c a od, f a (0, ) (, ), ad by c a od, f a (, ) (, ). Proof () Obvousy, he above appg aps A o B. The appg s oo, sce ca be reversed for a b B. () Frsy, we reark ha a eees of A, are aso eees of U. Ne, we ca easy verfy ha c for a a A. Moreover, sce ( a, ) =, we aso have ha ( c, ). Coversey, f c s odd ad ( c, ), we ca fd a a A wh c a or c a od, ad such ha a s he correspodg subse of (0, ). So, here are o oher eees U ha hose whch are AC. Lea. Le be a odd eger. () he reao a a defes a oe-o-oe appg fro he subse A of odd egers o he subse B of eve egers; U () he appg fro U o U defed by, f, ad, f, s a soorphs. () f U s a cycc group, he so s U, ad vce versa; (v) U ad U are boh cycc, for a odd pre p. p p

15 Proof () Fro (, a) foows ha (, a). Moreover, sce a s odd, a s eve. The appg s verbe, so s oe-o-oe. () We kow aready, by Lea., ha for U ad, we aso have U ad ha s odd. I he group bewee eees of U U U producs are ake od. I foows ha equaes are aso rue whe ake od, or saed equvaey, hey hod as equaes. Sce he appg s verbe, s a soorphs. () Ths foows edaey fro (). (v) For a pre p, we have ha U s he upcave group of he fed GF( p), ad p hs group has geerag eees. If s o (wce) a pre, he queso of U beg cycc or o s soewha ore cope. Le us frs cosder a group U k wh ( k, ). The eees of Z are perued aog each oher whe we add a fed eger o a hese eees. The sae hods whe we upy a eees by a fed eger whch s pre o. Therefore, a se { r k0 } coas egers pre o, for ay r. If we ake r such ha ( r, k), hese egers are aso pre o k. I foows easy ha( k) ( k) ( ) or equvaey U =U U, whch s of course a we-kow resu. More k k parcuar, we have U k U k U whch ca be proved by showg ha here s a soorphc appg bewee he groups a he hs ad a he rhs of hs reao. Le U k { a, a,..., a ( k )} ad U { b, b,..., b ( )}. The syse of cogrueces c a od k, c b od, has a uque souo c wh 0 k, for ay par, j, j j, j c, j of reeva dces, j, accordg o he Chese Reader Theore. I easy foows ha he group { 0 ( k), 0 j ( )} s soorphcay apped o U k c, j U k U by he appg c, j ab j. Eedg ad geerazg he above argues gves rse o he foowg properes. Lea. () If p... p, he U U U... U ; () U a C C a, for a ; () s cycc f ad oy f equas,, U (v) Le g be a geeraor of s a geeraor of a geeraor of U p U p U p p p a p or p a, for ay odd pre p; wh p odd, he a eas oe of he egers g ad g + p ad g s aso a geeraor of whe p s eve; U p whe g s odd, whereas g + p s

16 (v) I he eger geeraes he subgroup C. U a a The deas of he proofs ca be foud [6, Seco ]. Sce subgroups of a cycc group are cycc as we, ad by appyg Lea. (), we ow have he foowg geerazao of Theore.. Theore. If s equa o,, p or p,wh p a odd pre, he group K of (.9) wh de wh respec o G, s deca o he group G of -powers of G. Eape. Oe ca easy verfy he varous properes of Leas.. for 9. The we have U9 {,,,,7,8} ad U8 {,,7,,,7}. A geeraor of s, sce s frs s powers are,,, 8, 7,. The correspodg U 8 U 9 geeraor of s (= +9) (cf. Lea. (v)), he frs s powers of whch are,,, 7, 7,. Aoher eape s 6, wh U6 {,,,7,9,,,}. Ideed, oe ca edaey verfy ha U6 CC, where C {,,9,} s a cycc group of order, ad C {,7}. Eape. Take ad q. Sce s a pre, U {,,..., } s a cycc group. Furherore, we have od, ha, 9 ad, ad so r ord, H {,, 9} ad s /. We choose K : H, whch pes s ad. I appears ha he vaues of g, g U, are,, ad 9. So, we have deed K U. Ne, we eed K by addg he eees of K, so ow K : {,,,9,0,}, pyg s,. Deerg he squares of a eees of U shows ha K U. Sary, we ake q, whch subsequey gves r ord () =, H {,, 8,. } Choosg K : H, ad so s,. Aga, oe ca easy verfy ha K U. I he e eape s show, ha whe U s o cycc, he cuso K s o ecessary a equay. Eape. Take ad q. I foows ha r = ord () = ad H U, } U {. The group {,, 7, } C C, whch s o a cycc group. Is s edaey cear ha

17 s /. Choosg K H pes s ad. Deerg he squares of he eees of U shows ha U {} K. Sar resus are obaed for q 7 ad q. Eape. A arger eape s obaed by akg. The group U coas he egers {,,,7,8,,,,6,7,9,,,,, 7, 8,,,, 6, 7, 9, }. So, U=. Fro Lea. foows ha G : U U9U C 6 C, ad ha G s o cycc. Take furherore q 8, whch yeds H {,8,7,9} ad r, so s 6. 6 Choosg K : H gves 6 ad s. Now, G {,9}, whch s a proper subgroup of K. So, hs case K coas ore eees ha jus he 6-resdues. However, f we choose K : {, 8,7,9,, 8, 7, 9}, we have. Now we fd G K, or K s precsey he group of -resdues. Reark The as case Eape. shows ha a group K, as defed.6, ca be deca o G eve whe U s o cycc. The queso whe a eger a s a -resdue ca geera be aswered a sasfacory way by he foowg ea. I hs ea he oo of prve roo s used. A eger a wh ( a, ) = s sad o be a prve roo od, f s he saes posve eger such ha of. a Lea. Le p... p, ad e a be soe eger wh ( a, ). = od. Oe aso says ha a s a prve roo / d () f possesses prve roos, he a s a - resdue od, f ad oy f a od, where d (, ), ad f a s a - resdue, he a od has eacy (, ) souos; () he eger a s a - resdue f ad oy f he syse of cogrueces, a od p,.., a od p has a eas oe souo; () f a s odd, ad, he a od a od has precsey oe souo for odd, whereas for s eve, here are precsey d souos wh d (, ) f a od, / d a = od, ad here are o souos oherwse. For proofs, we refer o [6, Seco ]. As a cosequece of he prevous resus, we ca prove he foowg propery for he subgroup U cossg of he -powers of he eees of U. 6

18 Theore. () Le p... p, ad e be soe posve eger. The he order of he group U s equa o /(, ( ))(, ( p ))...(, ( p )), ecep whe ad s eve. () If ad s eve, hs order equas / (, )(, ( p ))...(, ( p )). Proof () We frs prove par () of he Lea for he case ha s a pre power p, where f p. Fro Lea. () ad Lea. () we kow ha hs case s cycc ad ha f a s a -resdue od, here are eacy (, ) eees U whch sasfy a od. Hece, U = /(, ). Ne, we assue for he sake of coveece ha s he produc of wo pre powers, say p p wh f p. We kow ha U U U. I foows ha s soorphc o he drec p p U p U p produc of he subgroups ad. Accordg o he frs par of he proof hese subgroups have order ( p ) /(, ( p )) ad ( p ) /(, ( p ), respecvey. Sce ( p, p ), we have ha ( p ) ( p ) ( ), ad so = /(, ( p ))(, ( p )). U ( ) / d d U U ) U () I he case ha ad eve, we appy he secod par of Lea. (). Now, =, wh (, ), ad he resu for foows edaey. U Eape. Take 6. We have ha U {,,7,,,7,,, 7,,, 7} ad 6 6 U {,7,, 7} {,9,, 9}. Sce ( 6), ad ( 9) 6, 6 Theore. () yeds U = /(,)(,6), whch s correc. A sar cacuao for gves respecvey U {,,7,,,7,9,}, U {} ad, by appyg Theore. (), U = () / (,)(,) 8/.. =. 7

19 Coroary. Le I U / U be he de of he group U wh respec o, : U. () For p for soe odd pre p ad for p, oe has ha f p he I,. () For, oe has ha f,, he I, for, I, for ad I, / for. Proof () Sce ( p ) p ( p ), (, ) (, ) I p p p. for ay pre p, foows fro Theore. ha ( p ), () Now we appy. The resu for s obaed fro Theore. () ad for fro Theore. (). I w be obvous ha Coroary. s equvae o Theore... Soe spe properes of geerazed resdue codes Le be a prve -h roo of uy yg soe eeso fed of F GF(q). Le R be a subse of Z whch s cosed uder upcao by q, ad e j be soe eger copre wh. Furherore, we roduce poyoas g ( ) ad j g ˆ( ) ( ) = ( ), for whch we ca easy prove he foowg propery R jr (cf. aso [, Theore.8]). Lea. Le c ( ) c ˆ( ) 0 0 c R be a poyoa wh coeffces F ad degree ess ha, ad e c ˆ, where cˆ c wh sj od, ad (j, ) =. The g () dvdes c() f ad oy f g ˆ( ) dvdes cˆ. s Proof ' Le j ' be he verse of j G,.e. jj od. The we have he foowg seres of equvaeces: g () c() c( ) = 0, R 8

20 jj' c( ) 0, R j' c( ) 0, jr cˆ ( ) 0, jr g ˆ( ) cˆ j C, q, ) Now, a poyoa c() s he code, f ad oy f g j ( c(). ( j) Sce g ( ) = ( ), foows fro he defo pror o Lea. ha jk gˆ g j K j, by akg R : K ad : j (cf. (.)). The e heore s a edae cosequece of ha ea. Theore. The codes,,a have deso /. Moreover, hey are equvae, C, q, ad hece hey have he sae u dsace. Proof The oc poyoa g s he produc of s dffere rreducbe poyoas over F (cf. (.)). Therefore, s he uque geeraor of a degree of he code ( g ). Hece, he deso of hs code s rs ( ) /. y j g Le c be a poyoa wh ozero wegh d, for every wh. These poyoas have he foowg properes (cf. aso [, Seco.]). Theore. () c 0 od (); () () P c r( ) 0 for soe a,, ad r 0 oherwse; P( ) c r( ) od ; (v) P c r( ) od. 0 r F( ), where r 0, f c 0 for () Proof 9

21 () For each he poyoa c s a upe of. The equay ow foows edaey fro (.). () Mupyg he equay () by P() (cf. eq. (.)), we oba or equvaey, wh P c 0 P c r( ) 0 od, 0, g r F[ ]. Fro eq. (.) we have P( ), hece P 0. If we assue () ha c 0 for a, he r 0, whereas r 0 whe c 0 for a eas oe - vaue. () Ths reao foows by dvdg hs ad rhs of he equay () by. (v) Fro () foows ha P( ) 0 od. We cobe hs cogruece wh he oe (). We ca easy verfy ha = r() od. Accordg o he Chese reader heore we ow ay cocude ha 0 c = r od. 0 0 () P( ) c Theore. For ay se of fed vaues for, q ad, he foowg reaos hod:, q, () C ( ) ; () for Proof, oe has C, q, R. () Le c C, q,. The we ca wre c( ) a( ) g = a () () by, q, (.). Hece, C ( ). Equay foows by reversg he above argue. () Frs we observe ha d ( j j j C C ) = d C + d C d C C = ( rs) ( rs) ( rs) pes saee (). j So, C C R for a, j {,,..., } ad j. Ths 0

22 Eape. Take ad q. So, G = U {,,..., }. I Eape. we aready derved r ad H {,, 9}. Furherore, ad so. The boa has he foowg decoposo GF[ ] ( ) ( )(... ) = ( )( )( )( )( ). The as four facors are rreducbe poyoas over GF() ad ca be defed wh poyoas, F (), {,,, } (cf. (.)). Ne, we ake K H, ad hus G/K =. Cosequey we ca wre G K K K K, K {,, 9}, K {,, 6}, K {, 0, }, K { 7, 8, }. Sce = s a pre we have accordg o Theore. or Theore. ha K s deca o U, as oe ca easy verfy. Le be a prve -h roo of uy soe eeso fed of GF(). The we defe (cf. (.)) g 9 K ( ) ( )( )( ) 9 = ( ) ( ). 9 Now, we pu : for whch we edaey ca see ha, ad so { 0,, }. Of course hs s cear fro he begg, sce he coeffces of are eees of GF(). I order o deere he vaues of he coeffces, we assue ha s defed as a zero of he rreducbe poyoa 0 g g whch has epoe. By sraghforward cacuao we ge,,, 0 g 9. I a copeey sar way we fd g, g ad g. The four codes geeraed by hese poyoas are erary -resdue codes accordg o Defo.. 9,, 0,, ad hece 0

23 . Mu dsaces geerazed resdue codes A edae cosequece of Theore. (v) s he foowg resu whch provdes us wh a ower boud for he u dsace d of codewords c() of a GR- code whch have he propery c(). I he e we cosder poyoas () c C, q, of wegh d (o ecessary he u wegh), ad such ha c 0 so ha we ca appy Theore.. Theore. Le d be he wegh of a poyoa C (), ad such ha c 0. If d s c, q, he wegh of he poyoa P(), he d d. P Proof Take c C of wegh d ad wh c 0. By suabe peruaos of s coeffces, usg he cosruco he proof of Lea., we ca rasfor hs poyoa o poyoas c,..., c a of wegh d ad whch aso sasfy () c 0,. As a cosequece of Theore. (v), he produc c s a ozero upe of.... Sce hs poyoa has wegh ad sce c has a os d ozero coeffces, he equay ow foows. The above heore ca be cosdered as a geerazao of a we-kow resu for,.e. for QR- codes ad aso for GQR-codes ad oher geerazaos. See aso [,,] ad [9, Theore 6.9.]. P I case ha s o a eee K, we ca eve derve a sroger resu. Theore. Le d be he wegh of a poyoa C () wh c 0. If K, he d P ( d d ) /. c, q, Proof If K, he beogs o a cose dffere fro K ( K). We sha deoe hs cose by K. Le ag be eher K or K, so a defes a cose K a dffere fro ad K. If ak, he a ak for soe k K, ad hece k whch K a ak K K s fase. So, K ad := are wo coses dffere fro ad. Coug a K a

24 a sar way shows ha G / K cosss of pars of coses K ad K for / dffere vaues. I aso foows ha K K, K, K, K,..., K /, K / K. I he e we sha use he abeg for he coses of K. The correspodg poyoas () ( (.) are deoed by g, g ) () (, g, g ) ( /,..., g ) ( /, g ). Take soe fed vaue. We ca wre rs rs K g ( ) ( ) ( ) ( ) K K K rs rs K = b g ( ), wh b ( ). Sce he coeffces of he poyoas g are eees of F, b us be a eee of F as we. We ca aso prove hs sraghforwardy by akg use of he varace of K uder upcao by q. I parcuar we ca wre q K K Hece, 0 q( q K r r j... s ) q (... s ) j0 j0 K ad cosequey g ad g gves edaey ha b = g ( 0 ). q j K K. F. Coparg he coeffces of Ne, we cosder c a g, beg a poyoa or codeword of C of e wegh d ad of degree e. The c c (/ ) a g, wh e a : a ( ), s a codeword of C wh he sae wegh d. The poyoa c c sads for a codeword he erseco code C C, whch cao be he zeroword, sce s o dvsbe by. So, has a posve wegh whch s a os d d. We ca do hs for a vaues of, /, sce a codes C are equvae accordg o Theore., ad so hey a have a codeword of wegh d. More geeray, he poyoa c c s he erseco code C C ad has wegh a os ( d d / ) / /. The equay ow foows fro Theore. (v). Theore. ca be see as a geerazao of a resu of Maso ad Sooo for quadrac resdue codes. See aso refs. [,9,,]. Fro Theores. ad. appears ha for deerg he u dsace of a GR- code, s essea o dsgush bewee codewords c () wh c 0 ad codewords wh C, q, C, q, c = 0. To hs ed, we roduced Defo. he subcode, geeraed by he poyoa ( ) g,, whch s caed he C, q, epurgaed code, whereas sef s caed he augeed code (cf. [, Seco.].

25 Corroary. If he u dsace d of he geerazed resdue code s odd, he sasfes () d d ; P / () d ( d d ), f K. P C, q, I w be cear ha Theores. ad. are geerazaos of he ower bouds for he u weghs of words whch are a augeed quadrac resdue code, bu o he epurgaed oe (cf. e.g. [,9,0,]), whch are obaed by akg. Furherore, sce GR-codes are cycc, we aways ca fd poyoas c, such ha () c ( 0) 0 for a By epog hs fac, we gh sghy prove he bouds of Theores. ad. he fuure. Reark I he proof of Theore. we showed ha a poyoa recproca poyoa of case ha g ( ), whe g s equa o he s o a eee of he group K. I he s a eee of K, here ess equay bewee a poyoa ad s ow recproca b rs g ( ). Ths ca be show by sar argues. We cocude ha he poyoas g, / /, are (a-)syerc wh respec o he vaues of her coeffces,.e. rs g b g ( ), b g ( 0). g Eape. Take 7 ad q. The G U 7 {,,,,, 6}. Oe easy verfes ha r = ord 7 () = ad H {,, }. Hece, sce ( 7) 6, we have. The oy possbe subgroup K of G wh H K G, s K H U 7, whch pes ha he resug codes are -resdue codes. So, K {,, } ad K {,, 6}. Le be a prve 7 h roo of uy soe eeso fed of GF(). Fro defo (.) we have g g ( )( )( ) ( )( )( ) 6 ( ( ) 6 ) ( ( 6 ), ). I order o deere he coeffces ore cosey, as eees of GF(), we pu 6 :. The foows ha 6( ). I urs ou ha sasfes he equay 0. Now, he equao 0 has hree

26 souos GF(),.e., ad 6. If we pu, we oba he poyoa whch s reducbe GF[ ], sce facorzes as ( )( ). Pug or 6 gves rse o he rreducbe poyoas ( ) ( 6 g ) ( ) 6 ad g. Ideed g g ( ), hus usrag he equay he proof of Theore.. Noce ha ( 6) K. Eape. Ne, we ake ad q 9. Sce 9 = od we have r. Furherore, G U {,,, } ad ( ), ad hece. The subgroup H = {, } of G has de, ad herefore we ake K H U. The poyoas correspodg o K {, } ad K {, } are respecvey g ( )( ) = ( ), g ( )( ) ( ), where s a prve h roo soe eeso fed of GF(9). The coeffce : sasfes he equay 0. The equao 0 has hree roos GF(9),.e., ad. Pug provdes us wh he reducbe poyoa = ( 8)( 7). Pug or gves rse o he rreducbe poyoas g ad g I hs case we have g g ( ) ad aso g g, usrag he reark rgh afer Theore.. Noce ha ow ( ) K. Eape. I Eape. we derved for he case ad q he poyoas g ad g g g ( ). Sce K. As oe ca verfy we have here ha, hs usraes he proof of Theore.. Sary, we have for g ad g he reao g g ( ). A four -resdue codes have he sae u dsace d, whch sasfes he equay d P ( d d ) accordg o Theore.. Sce P() s a cosa hs case, foows ha ( d d ) pyg d. Sce her geerag

27 ( ) poyoas g ad g have wegh, he codes C,, ad C,, ceary have u dsace whch shows ha hs uderboud s sharp. As a cosequece, ad aso have u dsace. I order o usrae C,, C,, hs by cosrucg geerag poyoas of wegh, we appy Lea. ad he cosruco used s proof. Le c : g C, ad ake j. The he verse j' od s equa o 7. Fro c, c, c 0 foows ĉ 8, ĉ 7, ĉ 0, respecvey, whe a oher coeffces of cˆ are equa o 0. Hece, 8 7 c ˆ. Accordg o Lea. hs poyoa of wegh geeraes he C code (reeber ha K ). Tha hs poyoa deed represes a eee of ( ) C : ( g ) = ( ) s deosraed by he facorzao = ( )( ). I a sar way, by akg j 7 ( K ), 6 ad hece j', we fd ha he poyoa s a geeraor of C. Oe ca 6 aso verfy ha ( )( ). For he sake of copeeess we aso cosder he case j ( ), j' 0, gvg rse o c ˆ = ( )( ), whch s a geeraor ( ) of wegh of C, dffere fro g. If we ake j K ), j' 9, he resu s ( 9 6 c ˆ = ( ) ( ), whch s aoher geeraor of wegh of C. K 8 7 Reark We kow fro eq. (.) ad Theore. ha he geerazed resdue codes, g C, q., have geeraor poyoas whch are a of he sae a degree rs. Eape. shows ha hese geeraors are o ecessary of he sae (a) wegh. By appyg he cosruco as used he proof of Lea., we w aways be abe o rasfor hese poyoas o geeraors of he sae (a) wegh, bu o ecessary of he sae degree. Ne, we cosder aga he geera case, ad we roduce he eger p p... p. Le u be he upcave order of he pre q oduo,.e. u s he eas posve u p p... p eger such ha q od. The we have over he fed F GF(q) f f... f, (.) wh u, ad where he poyoas f (),, are a of degree u, ad are rreducbe over F (cf. eqs. (.), (.) ad (.)). 6

28 Lea. / The poyoa f ( ),, s rreducbe over F f ad oy f r u /. Proof By he equay F F ( / ), ad by eqs. (.) ad (.), foows ha / / / ( )... F f( ) f ( )... f ( Because of he uqueess of he caoca facorzao of (), he poyoas f ( / ) are rreducbe f ad oy f. Sce a hese poyoas are of degree u /, ad sce he poyoas () are a of degree r, he Lea ow foows edaey. F Suppose he equay r u / hods. The we aso have. Therefore, f here ess a GR- code wh egh, here aso ess a GR- resdue code wh egh. Noce ha hs equay does o aways hod. For eape for q =, 8 = ad hece, we fd respecvey ( ) ( ), ( 8), r (sce 8 od 8) ad u (sce od ). Hece, u /. 8/ r. Now, eq.(.) gves, ad so, ad f. Ideed, he ). poyoa f ( ) ( )( ) s o rreducbe GF ()[ ]. A sar eape w be gve Eape.. Theore. If r u /, where r, u, ad are as defed above, he he u wegh of a geerazed resdue code wh egh s upperbouded by he u wegh of a geerazed resdue code of egh. The sae hods f oy words of odd wegh are cosdered boh codes. Proof / Accordg o Lea. we ca defy F f ( ). If he poyoas p,, are he geeraor poyoas of he GR-codes of egh, he / g p ( ) are he geerag poyoas of he GR- codes of egh. Therefore, f he poyoa c F[ ] of degree sasfes c() 0 od p /, he c ( ) 0 od g. Le d be he u wegh he code of egh. Assue ha here s a codeword c() he code of egh wh wegh / w d. The c( ) s a codeword he code of egh. Sce hs codeword aso has wegh w, we have a coradco. Ths proves he frs saee. The secod 7

29 / saee foows fro he fac ha for, he poyoas c () ad c( ) have he sae vaue. Eape. Le 6 ad q. Sce od 6, foows easy ha r ord 6 () =. For he cycooc poyoa 6 we fd GF[ ] he facorzao o rreducbe poyoas over GF() ( ) ( )( ). Fro r ad ( 6) 8, foows ha whch s accordace wh he above facorzao. Furherore, we fd G {,,, 7, 9,,, } ad H {,, 9, }. Hece, we ca oy ake K : H as a proper subgroup of G. So, G / H=, s, ad G H H H H. Accordg o defo (.), he geerag poyoas g ad g are equa o ( ) ad ( ), respecvey, H H where s a prve 6 h roo of uy soe eeso fed of GF(). Oe ca 9 verfy ha g ( )( )( ( ) equas eher or, depedg o he choce of. To hs ed oe has o oce ha, ad ha he coeffce of he rhs s equa o ( ) = 6 ( ), whe sasfes. Furherore, we have () = whch gves u. We cao appy Theore., sce 6, r,, ad u do o ee he codo of ha heore. 0 8 Eape. Take ad q 7. The G U {,, 7, } ad H {, 7}. The cycooc poyoa for equas (). Sce 7 od, we have r ad / r /. We choose K : H ad so. For he geerag fucos of he quadrac resdue codes we fd hs case 7 g ( )( ) ad g ( )( ), where s a prve h roo of uy soe eeso fed of GF (7). By sar argues as apped prevous eapes we fd g ad g. Sce K, we ca appy Theore.. Therefore, we copue P /( )

30 So, d P 8, 8( d d ), ad hece we oba he rva resu d, where d s he wegh of a codeword c( ) wh c() 0. Ne, we sha ry o derve a upperboud for d, usg Theore.. Sce, we cosder. = 6. The order of 7 od 6 s equa o u, ad herefore u / r ad so he codo of Theore. s sasfed. The cycooc poyoa for 6 s equa o ( )( ) f f (cf. 6 ( Eape.). The u wegh of he code of egh 6, geeraed by has wegh, sce f ( f ) sef s a codeword. f ) ( f ) () () The codes geeraed by g f( ) ad g f ( ) obvousy have u wegh. Sce, Theore. s rvay rue hs case.. The speca case p I Theore. ad Theore. was saed for -vaues,, p ad p wh p a odd pre, ha f a group K, as defed (.9), has de wh respec o G U, he K s deca o he group U of -resdues, or saed equvaey : I,. (.) So, for hese -vaues he GR-codes beog o he subcass of -resdue codes, for a whch dvde () (cf. he Iroduco). I hs seco ad he e wo secos we sha vesgae hese cases soewha coser. As a frs speca case of he geera heory, we pu equa o a sge pre,.e. p, whch eas ha he codes are of pre egh p. For, we w ge he cassca quadrac resdue codes (QR-codes). Frs, we ake for q a pre (power) dffere fro p. Acuay, we ca ake for q ay pre (power) whch s U. I hs case we have G U p {,,..., p}, (.) p G = ( p) p, (.) p p p..., (.) whe he poyoa P(), defed (.), s equa o he cosa. Lke Seco, we assue ha q has order r od p, ad ha H {, q, q,..., q r }. (.) 9

31 p Sce ( p, q), we ceray have q od p, ad hece r p. For he vara subgroup K of G, we ca ake he group U of squares od p. I s we kow ha he order of hs group, usuay deoed by Q, s equa o ( p) /, ad ha s oy cose, cossg of a osquares ad deoed by N, has he sae order. Ths foows edaey fro he fac ha f g s a geeraor of he upcave group of GF( p), he eve powers of g are squares, whereas he odd oes are osquares. Hece, ( p)/ f we choose q such ha s a quadrac resdue od p (square), we ge q od p, ad so r ( p ) / Therefore, whe q s chose o be a quadrac resdue, he group H of (.) s eher he copee group Q of -resdues or oe of s subgroups. Whe H Q, ca aways be eeded by aoher quadrac resdue, such ha we oba H K = Q. I he foows ha, s, ad we ca wre (cf. (.)) p K y K, (.6) G K H H... H s, H : H. (.7) The cose y K (.6) s he se of osquares od p G. I a ore coveoa oao, usg Q ad N, reao (.6) s wre as G Q N, (.8) wh Q : K ad N : yk. So, for ay p oe ca cosruc quadrac resdue codes of egh p, by choosg a approprae vaue for q. I parcuar here es bary quadrac resdue codes f p ± od 8. For such p-vaues, s a quadrac resdue od p. Eape. Le p 7 ad q. I foows ha r ad H {, q, q } {,,}. 6 Furherore, G {,,,,, 6}, 7... ad ( 7) 6. Hece, 6/. We defe K : H Q {,, }, wh N {,, 6} as s oy cose, ad so. I parcuar we ca wre e.g. ad, sce s /, G K K, For he poyoas g we fd K H. 0

32 g g Q N ( ) = ( )( )( ), ( ) ( )( )( ) 6, where s a prve 7 h roo of GF (reeber ha r ). The codes () () geeraed by g ad by are he wo equvae [7,, ] Hag codes. g Eape. Ne we ake p ad q. I hs case we have r. Furherore, G {,,, },, ( ), whe he group of squares od s equa o Q {, }. However, H {,,,} s o a proper subgroup of G, due o he fac ha s o a square hs case. The oy group K wh H K G, s K H G. Now, /, ad hece s. I hs rva case we fd g K ( ). g The - resdue code geeraed by s he bary repeo code [,, ]. Fro he above eape we ay cocude ha for r p,.e. whe q geeraes he group G, we aways w fd he rva repeo code wh paraeers [ p,, p ]. Quadrac resdue codes (QR-codes) are defed he eraure (cf. e.g. [,]) as cycc codes geeraed by he poyoas g ( ) ad g ( ). As we saw, a addoa requree for bary quadrac codes s ha has o be a quadrac resdue sef (cf. he frs es of hs seco). Ths s rue f ad oy f p ± od 8. If p ± od 8, he Q whch pes ha geeraes he copee group G,.e. H G. Ths s precsey he case for he - resdue codes, as usraed by Eape.. I order o sudy he case p ± od 8 ore cosey, we cosder he eape p. Eape. 0 9 For we have G : U {,,...,0}, ( )..., () 0. p Q I hs case s a geeraor of U, ad herefore he group K : H has de wh respec o G, ad he oy resug code s he rva [,, ] repeo code as aready aouced above. N

33 Ne we ake q whch yeds H {,,,9,}. The de of hs group equas, ad cosss of a -resdues (cf. Theore.). So, f we ake K : H, he () () poyoas g ( ) ad g ( ) are he geeraors of wo K C,, K C,, equvae quaerary -resdue codes ad. Here, s a prve -h roo of uy soe approprae eeso fed of GF (). Epc cacuaos show g g () () where s defed by 0. Oe coud aso ake,, q. Aga we fd H {,,9,, }, ad so by defg K : H, C,, we oba wo ore quadrac resdue codes, hs e over GF(),.e. ad. The defg rreducbe geerag poyoas are respecvey (cf. [, p. 8]) g g () (),, C,, These codes are equvae versos of he we-kow perfec erary Goay code wh paraeers [, 6, ]. So, he u dsace d s equa o, whch s oe ore ha he ower boud for dsaces he epurgaed code, as foows fro Theores. ad. (cf. aso Theore.). 8 Of course, oe coud aso ake q-vaues wh ( q,) whch are o U. E.g. q s equa o oduo. So, oduo, hs q-vaue geeraes he group H {,,9,, } whch aga s deca wh he group of quadrac resdues. Sce r ad hece 0 /, foows fro eq. (.) ha has a facorzao o wo rreducbe poyoas over 8 GF G of degree. These poyoas gve rse o wo 8 equvae quadrac resdue codes of egh over he fed GF, deoed by C 8, {, }. (Cf. aso Eape..),, Rearks If oe ca prove ha he u dsace of G s odd (ke Pess dd [] for bary QR-codes), he Theore. gves edaey d. By akg ad q, oe ca cosruc he bary Goay code of egh as a quadrac resdue code (cf. [, p. 8]).

34 For p he cycooc poyoa p () s equa o, ad so he poyoa P() (.) s deca o. As a edae cosequece of Theores. ad., we have he foowg resus for he u wegh d for codewords of a - resdue code whch have odd wegh. Theore. The wegh d of a codeword c of a - resdue code (>) of egh p for whch () c ( d d) / 0, sasfes d p. If p. K p, hs equay ca be sregheed o These resus ceary geeraze he case for quadrac resdue codes. Noce ha Q, f ad oy f p od. 6. The speca case p As a secod speca case we ake p, wh p a odd pre, ad where q s aga a pre power wh ( p, q). So, we ca choose for q ay pre power whch, oduo p p, s U. For we he ge he geerazed quadrac resdue codes (GQRcodes) as dea wh [0]. We ow have G U {,,..., p }\{ p, p,..., p }, (6.) p G= ( p ) p ( p ), (6.) ( p) p ( p) p... = p p p /. (6.) For he poyoa P defed eq. (.) we fd p p p P /.... (6.) Jus ke Seco, we ake for q soe pre power wh ( p, q). We assue ha q has order r oduo p, ad we roduce he subgroup H of G by H q q q r {,,,..., }. (6.) Sce we kow ha q ( p ) od p, foows ha r ( p ).

35 Lke he subcase, whch was cosdered he prevous seco, he group s cycc (cf. Lea. ()). Le g be a geeraor of hs group. The he eve powers of g for he subgroup Q of squares of order ( p ) /. The odd powers for s oy cose N coag he osquares ad whch has he sae sze. Hece, f we choose q such ha s a quadrac resdue od p, we ge q ( p )/ od p, ad so r ( p ) /. Thus hs case he group H of (6.) s eher he group Q of -resdues or oe of s subgroups. Whe H Q ca aways be eeded o a group K by aoher quadrac resdue such ha we oba H K Q. Jus ke Seco, ow foows ha, s, ad he reaos (.6), (.7) ad (.8) aso hod hs case. So, for ay pre p ad for ay posve eger we ca cosruc quadrac resdue codes by choosg a approprae vaue for q. Eape 6. If we ge U9 {,,,,7,8} ad (9) 6. For q we oba he subgroup K : H {,,,8,7,} whch has de. The correspodg -resdue code s a bary code of egh 9 ad deso. Lkewse we derve for oher eees of : q K : H {,,7} = U, r,, I9, (, (9)) (,6), 9 U 9 U p yeds wo equvae quaerary -resdue groups of egh 9 ad deso 6; q K : H {,,7,8,, } = U9, r 6,, I9, (,6), yeds oe -ary code of egh 9 ad deso ; q 7 K : H {,7, } = U, r,, I9, (,6), 9 yeds wo 7-ary quadrac resdue codes of egh 9 ad deso 6; q 8 K : H {,8 } = U, r,, I9, (,6), 9 yeds hree 8-ary -resdue codes of egh 9 ad deso 7. Eape 6. For 7 we fd U 7 {,,...,6} \{,6,..., }, ad (7) 8. Sce s a geeraor of, he code C based o he group K : H U 7 7,, s a bary code of egh 7 ad deso 9. Sary, akg q yeds a -ary code of egh 7 ad deso 9. The choce q gves H {,,6,0,,,9,,7} = U. So, defg K : H provdes us wh wo equvae quaerary quadrac resdue codes C, {, }, of deso ,,

36 Fay, we ake 6 q 0 od 7, gvg H {,0,9} U. So, I accordace wh Corroary. (). Defg K : H yeds s equvae 6-resdue 6 codes over GF. Sce oe ca aso wre 0 od 7, here are aso s of such 6 7 7,6 6 codes over GF( ). I foows fro eq. (6.) ha he wegh of he poyoa P, defed (.), s equa o p. Subsug dp p Theores. ad. shows ha he equay of Theore. aso hods for. Theore 6. The wegh d of a codeword ( c ) ( ) of a -resdue code (>) of egh p for whch ( ) c () 0 sasfes d p. If K hs equay ca be sregheed o ( d d) / p. Eape 6. Cosder he quaerary -resdue codes C, {, } of Eape 6.. Fro Theore ( 6. foows ha he wegh d of a codeword ) ( c wh c ) () 0 sasfes 7,, d d. Noce, ha K. Hece, d. I order o appy Theore., we sudy quaerary resdue codes of egh. Now he order of od s equa o, ad sce 9/ = 7/, we are reay eed o appy. We facorze GF()[ ] accordg o ( )( ), where 0. The codeword c (or ) ceary has wegh, whe c() 0. So, d accordg o Theore., ad herefore d. Of course, he u dsace of he copee code C, {, } equas aso. To verfy hs, we facorze 7 / GF ()[ ] as he produc of wo poyoas of degree 9 as ( )( ). Ideed, he (geerag) codeword 9 7,, (or 9 ) s a word of wegh. 7. The speca case p The hrd speca case s whe foowg reaos hod p, wh p soe odd pre. I hs case he G U {,,..., p }\ { p,p,...,( p) p }, (7.) p G = ( p ) p ( p), (7.) p p /. (7.) p

37 Sary as (6.), we ca wre for he poyoa eq. (.) P() p p p /.... (7.) Le q be soe pre power such ha ( p, q), ad assue ha q has order r oduo p. Le furherore H {, q, q,..., q r }. (7.) ( p ) Sce q od p, foows ha r ( p ). The group U s cycc (cf. Lea. ()), so here ess a eee g whch geeraes a eees of he group. The eve powers of g for he subgroup Q of squares of order ( p ) / ad he odd powers s oy cose N coag a osquares. So, f he chose pre power q s a quadrac resdue, s order r s a dvsor of ( p ) /. I ha case H s eher he group Q or a subgroup of Q whch ca be eeded o Q. Jus ke Secos ad 6, we he have, s, whe he reaos (.6), (.7) ad (.8) aso hod aga. The poyoa P (7.) has wegh p. Subsug hs vaue Theores. ad. yeds a resu sar o Theores (.) ad (6.). Theore 7. The wegh d of a codeword p ( c ) of a -resdue code (>) of egh p for whch ( ) c () 0, sasfes d p. If K, hs equay ca be sregheed o ( d d) / p. Eape 7. Le 0.. Successvey we ge U0 {,,7,9}, (0), U0 {,9}. Furherore, we ake q, gvg r,, ad K U0. The correspodg code s a erary code of egh 0 ad deso 6. If we ake q 9 foows ha r ad. We defe K : H {,9} U0 ad hece. The wo resug codes are quadrac resdue codes over GF(9). To deere he a geeraors of hese codes, we have o facorze GF(9)[ ] he cycooc poyoa 0 o wo rreducbe poyoas of degree. We ca do hs by wrg 0 = ( ), where s defed as a zero of he rreducbe (over GF() ) poyoa. 6

38 The poyoas he rgh had sde are he a geeraors g () ad g () (cf. Seco ) of wo equvae quadrac resdue codes over GF (9) of egh 0 ad deso 8. Eape 7. Take 8. (cf. aso Eape.). We ge U8 {,,7,,,7}, (8) 6. Ne we ake q 7 ad K : H {,7,}. Hece, r, ad. We facorze 9 o wo rreducbe poyoas of degree GF(7) [] accordg o 6 8 ( )( ). I foows edaey ha here are wo quadrac resdue codes over GF (7) of deso wh u wegh (dsace). Theore U 8 7. yeds hs case d d. The u wegh of he wo codes s he eas vaue sasfyg hs equay. 8. Eeded geerazed resdue codes ad he case = I hs seco we sha prese a eapes of a eeded bary GR-codes for =. Ths eape aso shows ha a GR-code wh s o aways a -resdue code. Defo 8. The eeded geerazed resdue code s he code obaed by appedg a overa pary-check o he (augeed) geerazed resdue code of egh over GF (q. C, q, ) Eape 8. Take ad q,.e. F : GF() {0,,, }, where s a zero of GF()[ ]. I foows ha G = U {,,,,8,0,,,6,7,9,0}, ad r : ord ()=. The group H (= =<> ad s coses U are H ) H {,,6}, H {,8, }, H {,7,0 }, H {0,,9 }. We defe a prve -h roo of uy he foowg way. Such a roo s a eee of GF( ), ad hece s a zero of soe rreducbe poyoa of degree over he fed F (ad o over GF() ). Le be a zero of. I foows ha 6 7,, ad. So, he order of he upcave group of GF () s, ad hece s a prve roo of uy. By soe spe cacuaos we fd 0,, ad. These reaos eabe us o derve ha he rreducbe poyoas over whch correspod o he coses,, ad H are respecvey P, P, P ad 6 F H H H 7 0 7

39 P. A aerave approach s o verfy ha, ad are zeros of P (), P () ad P (), respecvey. We cocude ha P P P P s he facorzao of he cycooc poyoa o rreducbe poyoas over F. Sce G : C6 C, here are hree possbe choces for a group K sasfyg H K G (cf. (.9)). We oba hese groups K ', K '' ad K ''' by eedg H wh he eees of K, K or K, The hree groups are a cycc of order 6, ad hece of de, whe hey are geeraed by, ad 0, respecvey. Correspodg o hese hree cases, we have he foowg g - fucos cf. eqs. (.) ad (.)): g P P, () 6 g P P, () 6 U 0 () 6 g P P, () 6 g P P, () 6 g P P, 6 g P P. () g g The poyoas ad geerae wo GR-codes over GF (), correspodg o he groups K ', K '' ad K '''. Oe ca verfy ha U {,,6} K', K '', K '''. So, he GR-codes correspodg o hese groups are o -resdue codes. () () Sce g ad g are aso poyoas over GF (), he GR-codes geeraed by hese poyoas are eve bary codes. Acuay, hese codes ca aso be produced by akg q whch has order 6 od, ad by choosg K : H {,,,8,6,} whch s a subgroup of de. So, ad s. Hece, ( ) ca be facorzed o wo rreducbe poyoas over GF (). I fac we have for a {,,} (cf. eq. (.)) ( ) g g ad P. I order o deere he depoe poyoas of he above codes, we sha appy he foowg resu eoed [, p.]. 8

40 Theore 8. Le C be a cycc code of egh, wh odd. Le g() be he geeraor of C of a degree ad e h () be s check poyoa. If r() s he recproca poyoa of h (), he s depoe geeraor e() s equa o where r '( ) s he dervave of r(). We reark ha we ca aso wre '(/ ), deg( h g r e g s, (8.) where s () s he recproca poyoa of r'.,, To appy (8.) o he code C, geeraed by, we derve successvey ( h ) () () e g ( ) = ( ) ( )., r, 0 6 s, ad fay I a sar way we fd he depoes 9 8 g () () ( ) ( ) e g ( ) =. e g ( ) = () () ( 0 7 ) ( ( ) ( ). () () 8 e g ( ( ) ) = 9 0 ) ( 8 9 ) ( e g ( ) = () () ( ( ( ) ( 9 0 ) ( ( ) ( ) ) ( ) ( ) ) ( 6 ) ( 8 9 ) +. 6 ) ( 6 ) + ) 9

41 e g ( ) = () () 8 ( 0 ( ) ( ) ( ) ) ( 6 ) ( 8 ) +. As oe ca see, a cases we have ha he se of epoes of he ozero -powers s a uo of cycooc coses od, wh respec o, where dcaes he sze of he fed GF (). Le us cosder he bary cycc -resdue code C,, of egh, whch s geeraed () () by g or by e. Accordg o Corroary., f he u dsace d s odd, he d d. Hece, we ed up wh he rva equay d. I s easy o 9 check ha he code does o coa words of wegh. Furherore, sce () 7 7 e ( ) =, he poyoa he rhs s a codeword accordg o a we-kow propery of depoes, ad so he code coas a word of wegh. Therefore, he u dsace d equas. The poyoa () e gves rse o he ( ) - geeraor ar ( ) c, where he subscrp c sads for cycc shfs over oe poso o he rgh. We ca rearrage hs ar accordg o cycooc casses he foowg way. Rows ad cous are abeed respecvey by,, 6;, 8, ;, 0, 7; 0, 9, ; 0; 7; ;,, 6; 9,, 8. The groups of dces separaed by se-coos, correspod o he cycooc casses. These ca be see as he orbs of he group U whe acg o he se Z. The ar sef ca be cosdered as a bock ar cossg of bocks ad bocks. These bocks heseves are crcuas. The ar epc for s o he e page. 0

42 Noce ha he frs four cycooc casses are he coses of he group H G. The eeded code ca easy be obaed by addg a addoa row ad cou o he above ar (abeed by ) cossg of jus oes. The word c GF() wh oes a posos

43 c 0, c0 c7 c, ad wh a oher coordaes equa o 0, beogs o he eeded C,, e code ad has wegh. The code has herefore u dsace. Reark The queso how o eed obary GR-codes s s dscusso (cf. aso [9, Sec. 6.9]). Refereces. E.R. Berekap, Agebrac Codg Theory, McGraw-H, New York, R.A. Bruad ad V.S. Pess, Poyadc Codes, Dscree App. Mah. (989), 7.. P. Cao, Goba Quadrac Abea Codes, Iforao Theory, CISM Courses ad Lecures 9, G. Logo (ed.), Sprger Verag, We, 97.. P. Desare, Majory Logc Decodabe Codes Derved fro Fe IversvePaes,If. ad Coro 8 (97), 9.. W.C. Huffa, The Auoorphs Groups of he Geerazed Quadrac Resdue Codes, IEEE Tras. If. Theory (99), K. Iread ad M. Rose, A Cassca Iroduco o Moder Nuber Theory, Graduae Tes Maheacs 8, Sprger Verag, New York, J.S. Leo, J.M. Masey ad V. Pess, Duadc Codes, IEEE Tras. If. Theory 0 (98), R.L. Ld ad H. Nederreer, Iroduco o Fe Feds ad her Appcaos (rev. ed.),cabrdge Uversy Press, Cabrdge J.H. va L, Codg Theory, Sprger Verag, New York, J.H. va L ad F.J. MacWas, Geerazed Quadrac Resdue Codes, IEEE Tras. If. Theory (978), J.H. va L, Geerazed Quadrac Resdue Codes, Lecure Noes, Dep. of Mah., Edhove Uv. of Techoogy.. F.J. MacWas ad N.J.A. Soae, The Theory of Error-Correcg Codes, Norh Hoad Pub. Copay, Aserda, V. Pess, Iroduco o he Theory of Error-Correcg Codes ( d ed.), Wey Iersc. Pub., Joh Wey ad Sos, New York, V. Pess, Q Codes, J. Cob. Theory, Ser. A (986), V. Pess ad J.J. Rushaa, Tradc Codes, Lear Agebrac App. 98 (988),.!6. M.H.M. Sd, Duadc Codes, IEEE Tras. If. Theory (987),. 7. A. Shara, G.K. Baksh ad M. Raka, Poyadc Codes of Pre Power Legh, Fe feds ad her Appcaos (007), H.N. Ward, Qudrac Resdue Codes ad Sypecc Groups, J. of Agebra 9 (97), 0 7.