Linear codes invariant with respect to generalized shift operators

Transcription

1 Liear codes ivaria wih respec o geeralized shif operaors V G Labues ad E Osheimer Ural Sae ores Egieerig Uiversiy Sibirsy ra 37 Eaeriburg Russia 6 Caprica LLC Pompao Beach lorida US Absrac The purpose of his paper is o iroduce ew liear codes wih geeralized symmery We exed cyclic ad group codes i he followig way We iroduce codes ivaria wih respec o a family of geeralized shif operaors (GSO I paricle case whe his family is a group (cyclic or Abelia hese codes are ordiary cyclic ad group codes They are ivaria wih respec o his group We deal wih GSO-ivaria codes wih fas code ad ecode procedures based o fas geeralized ourier rasforms The hope is ha hesemore geeral srucures will lead o larger classes of useful codes good properies Iroducio Le be a fiie field A bloc code of legh is a subse Cof ie a collecio of legh vecors wih compoes from Mos of he lieraure o bloc codes perais o bloc codes over s fiie fields = G ( q or fiie rigs = GR ( q where q= p ad p is a prime Alhough ay subse forms a code here are codes wih more srucure ha are very useful ad compose he majoriy of bloc codes i pracice A liear bloc code is a bloc code ha is a -subspace of he -vecor space I addiio o lieariy here are may srucural properies ha mae for good codes Oe of he mos prevale such srucural properies is symmery of code ha is described as ivariace wih respec o a group Ivariace (code symmery i may circumsaces leads o some ice ecodig ad decodig algorihms ye i is a very simple srucure o describe or hese reasos i is oe of he mos sudied srucural properies i codig heory Defiiio [] A cyclic bloc code C of legh over a fiie field is a liear bloc code wih he propery ha if ( c c c c Che ( c c c 3 c C I meas ha group of code symmery of a cyclic code C is Cyclic codes are sudied from may pois of view Oe way is o view hem as ideals of a algebra Defie ρ: [ x]/ x via ρ : ( c c c c c + cx + + c x + c x I ca be show ha ρ is a isomorphism Le C be cyclic bloc code The ρ( C is a subspace of he -vecor space [ x]/ x ow he added codiio of beig cyclic raslaes o he followig: if ρ( c ρ( C he x ρ ( c = x ( c + cx + + c x + c x = ( c + cx + cx + + c x ρ( C Wih his exra codiio ρ( C [ x]/ x There are may geeralizaios of cyclic codes some of which may be viewed as ideals of paricular rigs [3]: IV Ieraioal Coferece o "Iformaio Techology ad aoechology" (ITT-8

2 Daa Sciece V G Labues ad E Osheimer egacyclic (sew-cyclic codes[4-]- ideal of he rig A lg ( [ x] / x + cosacyclic codes []- ideal of he rig A lg ( [ x] / x λ where λ A lg( polycyclic codes [3]- ideal of he rig A lg ( [ x]/ f ( x where f ( x A lg( [ x] The ermiology of he cyclic codes heory may be exeded o defie a larger family ofcodes We sar by iroducig vecor-iduced clocwise ad couerclocwise shifs Give a vecor s= ( s s s s he s -clocwise ad s -couerclocwise shifs of codeword C= = ( c c c c are he followig correspodeces s s R c = R ( c c c = ( c c c + c ( s s s s = = ( c s c + sc c + sc c + s c s s Lc= L( c c c = ( c c c + c ( s s s s = = ( c + sc c + sc c + s c s c Dyadic codes are defied oly for legh a power of say = as follows i i= i i i i Deoe is radix- Defiiio or ay ieger { } le ( represeaio where l l l= i i i i i i = = ad Dyadic addiio of wo umbers i ad j deoed by i j is defied by ( ( = i j = i i i i j j j j = ( i j i j i j i j ( = = i for l = where = ( i j mod for l = The dyadic shif m= of a vecor l l l c c c is he vecor ( c m c m c( m ( Defiiio 3 Liear code of legh = is called dyadic code if he m -dyadic shif of every codeword is also a codeword for all m= The class of dyadic codes is a special case of abelia group codes [3 4-6] which is briefly discussed i he hird I his paper we would lie oiroduce ew liear codes wih geeralized symmery We exed cyclic ad group codes i he followig way We iroduce codes ivaria wih respec o a family of geeralized shif operaors (GSO I paricle case whe his family is a group (cyclic or Abelia hese codes are ordiary cyclic ad group codes They are ivaria wih respec o his group We deal wih GSO-ivaria codes wih fas code ad ecode procedures based o fas geeralized ourierrasformsthe hope is ha hese more geeral srucures will lead o larger classes of useful codes good properies The res of he paper isorgaized as follows: i Secio ad 3 he proposed mehod based o families of geeralized shif operaors (GSO is explaied Mehods Geeralized shif operaor The purpose of his subsecio is o iroduce he mahemaical represeaios of geeralized shif operaors associaed wih arbirary orhogoal (or uiary ourier rasforms ( -rasforms or illusraio we also paricularize our resuls for may rasforms popular i codig ad sigal heories The ordiary group shif operaors ( T f ( = f( + play he leadig role i all he properies ad ools of he ourier rasform meioed above I order o develop for each orhogoal rasform a similar wide se of ools ad properies as he ourier rasform has we associae a family of commuaive geeralized shif operaors (GSO wih each orhogoal (uiary rasform Such families form hypergroups I 934 Mary [78] ad HS Wall [9] idepedely iroduced IV Ieraioal Coferece o "Iformaio Techology ad aoechology" (ITT-8 339

3 Daa Sciece V G Labues ad E Osheimer he oio of hypergroup Oly i paricular cases hese families are Abelia groups ad hyperharmoic aalysis is he classical ourier harmoic aalysis o groups Le f( : Ω be a -valued sigal where be a fiie field Usually Ω= [ ] d i codig heory ad digial sigal processig where d is he dimesio of Ω : d = dim( Ω Le L( { f f } be vecor space of -valued fucios where card( Ω Ω : = ( (: Ω d Ω= Ω = The heory of geeralized shif operaors was iiiaed by Levia [] [] Accordig o Levia he family of geeralized shif T f(: = f( ( depedig o Ωas a parameer is defied i sigal space operaors (GSOs [ ] L( Ω by he followig axioms Axiom or all fucios f ( f ( L( holds Ω ad ay cosas ab he followig relaio ˆ [ ] ˆ [ ] ˆ T a f ( + b f ( = a T f ( + b T [ f( ] Axiom or a arbirary fucio f( L( Ω ad arbirary sr Ωi holds r r ( r r T T [ f( ] = T T [ f( ] or f ( ( r = f (( r ie T = T( i e he GSOs are associaive Axiom 3 There exiss a eleme ( ( ( (r (r ( ( T f ( f ( Ωwih [ ] f( L Ω This meas ha he family of GSOs coais ideiy operaor If moreover he followig axiom is fulfilled he he GSOs are called commuaive Axiom 4 or ay elemes f( L Ω holds Ωad arbirary ( [ ] [ ] ( ( ( ( x for all Ωad for all r r r r T T f( = T r T f( or f ( (r r = f ( r( r ie TT = TrT (3 We expad oio GSOs o he more complex sigal space Le f ( : Ω A lg( be a A lg( - valued sigal The se Ω of he values of he variable cosiues he domai of he sigal Usually Ω= [ ] d i codig heory ad digial sigal processig where d is he dimesio of Ω : d = dim( Ω The se A lg( of values of he sigal f( is he rage of he sigal Abou he rage of he sigal we assume ha A lg( is a commuaive algebra wih aivoluio operaio a a a A lg( I paricular if A lg( is he complex field he he ivoluio operaio is complex cojugae Le Ω be he space dual o Ω The firs oe will be called he specral domai he secod oe be called sigal domai eepig he origial oio of Ω as «ime» ad as «frequecy» Le ( Ω L Ω Alg( : = f ( f ( : Ω Alg( Alg ( ( { } L Ω Alg( : = ( ( : Ω Alg( Alg ( d be wo vecor spaces of A lg( -valued fucios Here Ω ϕ ( x be a Ω=Ω = Le orhoormal sysem of fucios i L( Ω A lg( The for ay fucio f ( L( Ω lg( exiss such a fucio ( L( Ω lg( - = ( = = ( = A for which he followig equaios hold: A here ( f ( f( ϕ ( f( ( ( ϕ ( Ω (4 ( L Ω A lg( is called he ourier specrum ( -specrum of he A lg( - The fucio ( valued sigal f ( L( Ω lg( A ad expressios (-( are called he pair of geeralized ourier rasforms (or -rasforms I he followig we will use he oaio f( ( i order o idicae -rasforms pair IV Ieraioal Coferece o "Iformaio Techology ad aoechology" (ITT-8 34

4 Daa Sciece V G Labues ad E Osheimer A fudameal ad impora ool of codig ad sigal heories are shif operaors i he «ime» ad «frequecy» domais They are defied as ( T f ( : = f( + ( D ( : = ( + ad ( T f ( : = f( ( D ( : = ( j or f( = e ad ( e j = we have j j( + j j λ ( j j j( j j = = = λ ( j j j + j j j Te = e = e e = e De = e = e e = λ( e ad (5 j j( j j j T e e e e e De = e = e e = λ( e ie harmoic sigals e j j ad e are eigefucios of «ime»-shif ad «frequecy»-shif operaors T T j j ad D D correspodig o eigevalues ( = e ( = e λ λ ad j j λ( = e λ ( = e respecively Defiiio 4 The followig operaors (wih respec o which all basis fucios are ivaria eigefucios T ϕ (: = ϕ ( ϕ ( = λ ( ϕ ( Ω ad ( ( ( T ϕ (: = ϕ ( ϕ ( = λ ( ϕ ( Ω ( ( D ϕ (: =ϕ( ϕ ( =λ( ϕ( Ω Dϕ (: =ϕ( ϕ ( =λ( ϕ( Ω (7 are called commuaive geeralized "ime" shif ad "frequecy" shif operaors (GSO s respecively where λ( =ϕ( λ( =ϕ( ad λ ( =ϕ( λ ( =ϕ( are eigevalues of GSO s T T ad D D respecively or hese operaors we iroduce he followig desigaios: T ϕ ( : =ϕ ( ( T ϕ ( : =ϕ ( ' Ω ( ( ( Dϕ =ϕ ( Dϕ =ϕ $ Ω (: ( (: ( here symbols ( ' $ deoe quasi-sums ad quasi-differeces respecively If T T ad D D are marix elemes of operaors α α T = T T = T σ σ ad D = D α α D = D α α he ad ( T ϕ ( =ϕ ( ( =ϕ ( ϕ ( = T ϕ ( σ ( σ σ Ω T ϕ ( =ϕ ( ' =ϕ ( ϕ ( = T ϕ ( σ ( ( σ σ σ Ω D ϕ ( =ϕ ( =ϕ ( ϕ ( = D ϕ ( α α α Ω D ϕ ( =ϕ ( =ϕ ( ϕ ( = D ϕ ( $ α α α Ω The expressios (8 (9 are called muliplicaio formulae for basis fucios ϕ ( L Ω lg( ϕ ( L Ω A lg ( They show ha he se of basis fucios ( A ad { } ( Ω form wo hypergroups wih respec o muliplicaio rules (8 ad (9 respecively Cosequely wo σ (6 σ (8 (9 IV Ieraioal Coferece o "Iformaio Techology ad aoechology" (ITT-8 34

5 Daa Sciece V G Labues ad E Osheimer spaces L( Ω A lg( ad L( Ω lg ( A form ime ad frequecy algebras wih srucure cosas T ad σ D respecively α rom (8 ad (9 we easily obai he marix elemes of he GSOs i ime ad frequecy domais T = ϕ ( ϕ ( ϕ ( σ T = ϕ ( ϕ ( ϕ ( σ ( σ σ ( D = ϕ ( ϕ ( ϕ ( D = ϕ ( ϕ ( ϕ ( α α α α Ω Ω The expressios ( ( ca be compacly wrie o he operaor laguage T = diag ϕ ( T = diag ϕ ( x { } { } D = diag { ϕ( } D = diag { ϕ( } where diag{ ϕ} deoes a diagoal marix which eries cosis of values of he fucio ϕ If here exis such eleme ha he equaio ϕ( for all is fulfilled he here exis he ideiy GSO i ime domai Ideed he subsiuio of io he expressios ( gives T = diag ϕ ( = = diag = = = = I { } diag ( = diag = = T = ϕ = = = I If here exis such a eleme ha he equaio ϕ ( x for all x Ωis fulfilled oo he here exis he ideiy GSO i frequecy domai Ideed he subsiuio of io he expressios ( gives ˆ D = diag ϕ ( x = diag = = I { } ˆ D = diag { ϕ ( x } = diag{ } = = I We see also ha wo families of ime ad frequecy GSOs form wo hypergroups HG= { T } = { D } ϕ( ad{ ϕ } HG By defiiio fucios Ω ( Ω Ω ( (3 ad are eigefucios of GSOs or his reaso we ca call hem hypercharacers of hypergroups or a sigal f ( L( Ω A lg( defie is shifed copies by f( ( = ( T f ( = T ( ϕ ( = ( ( T ϕ ( = ( = ( ϕ ( ϕ ( = ( ϕ ( ϕ ( f( ' = ( T f ( = T ( ϕ ( = ( ( T ϕ ( = ( = ( ϕ ( ϕ ( = ( ϕ ( ϕ ( Aalogously for a specrum ( L( Ω A lg( ( = ( D ( = D f( ϕ ( = f( ( Dϕ ( = ( = f( ϕ ( ϕ ( = f( ϕ ( ϕ ( ( $ = ( D ( = D f( ϕ ( = f( ( Dϕ ( = ( = f( ϕ ( ϕ ( = f( ϕ ( ϕ ( we (4 (5 IV Ieraioal Coferece o "Iformaio Techology ad aoechology" (ITT-8 34

6 Daa Sciece V G Labues ad E Osheimer We will eed i he followig modulaio operaors: M f (: =ϕ ( f( M f (: =ϕ ( f( ( ( ( ( M ( : =ϕ ( ( M ( : =ϕ ( ( rom he GSOs defiiio i follows he followig resul (wo heorems abou shifs ad modulaios Shifs ad modulaios are coeced as follows: f( ( ( ϕ ( f( ' ( ϕ ( ad ( T f ( M ( T f ( M ( ( ( ( ( f( ϕ ( ( $ f( ϕ ( ( ( ( ( D ( M f ( D ( M f ( Geeralized covoluios ad correlaios Usig he oio GSO we ca formally geeralize he defiiios of covoluio ad correlaio Defiiio 5 The followig fucios ad ( ( ' ( ( ( ( ( $ y (: = h x( = h( x Y : = H = H Ω Ω ( ( ' ( ( ( ( $ c( = f g(: = fg ( C( = G : = G Ω are called he - ad - covoluios ad he cross - ad - correlaio fucios respecively associaed wih a classical ourier rasform If f = g ad = G he cross correlaio fucios are called he - ad - auocorrelaio fucios L Ω lg( L Ω A lg ( equipped muliplicaios ad form The spaces ( A ad ( commuaive sigal ad specral covoluio algebras L( Ω lg respecively ( A ad L( Ω lg A ( Theorem Le us ae wo riples y( h( x( L( Ω A lg( ad y( h( x( L( Ω lg( Obviously Y( H( X( L( Ω A lg( ad Y( H ( X ( L( Ω A lg( Le ( ( ' ad Y( = ( H X ( = H( ( y ( = h x ( = h ( x Ω $ Ω A he geeralized ourier rasforms ad map -ad -covoluios io he producs of specra ad sigals respecively { y} = { h x} = { h} { x} { Y} : = { H X} = { H} { X} ie y( = ( h x ( Y( = H( X( y( = h( x( Y( = ( H X ( c ( f ( g ( L Ω lg( c ( f ( g ( L Ω A lg( ad Theorem Le us ae four riples ( A ( C( ( G( L( Ω A lg( C( ( G( L( Ω A lg( Le ( ( ' ad C = ( G( = ( G( c( = f g ( = f( g Ω ( $ he geeralized ourier rasforms ad map - ad -correlaios io he producs of specra ad sigals respecively { c} = { f g} = { f} { g} { C} : = { G} = { } { G} ie IV Ieraioal Coferece o "Iformaio Techology ad aoechology" (ITT-8 343

7 Daa Sciece V G Labues ad E Osheimer ( ( c f g C G c f g C G ( = ( ( = ( ( ( = ( ( ( = ( 3 Codes ivaria wih respec o GSOs We are goig o cosider bloc codes of legh as subses L( Ω lg( C L( Ω Alg ( ie a collecios of legh vecors wih compoes from ( { ϕ } ad { ϕ ( } be orhoormal sysems of fucios for L( Ω A lg( ad L( Ω lg ( ( Ω respecively They geerae wo hypergroups HG- ad HG Defiiio 6 HG- ad L Ω lg( C A ad Alg Le A HG - ivaria bloc codes C ( A ad L( Ω lg ( C A are liear bloc codes wih he propery ha if c ( Cad C( C he ( Tc ( = c ( ' C T HGad ( DC ( = C( $ C D HG respecively I meas ha HG- ad HG - ivaria bloc codes C L( Ω A lg( ad L( Ω lg ( C A have hypergroup symmeries Reed-Solomo (RS codes are obiary cyclic codes [3] The mos aural defiiio of HG- ad HG - ivaria RS codesare i erms of a cerai evaluaio maps from he subspace A lg ( of all -uples m = ( m m m (iformaio symbols = massage over A lg( o he se of C= = Cod A lg( L Ω A lg( codewords [ ] ( ( m= ( m m m c( = ( c( c( c( Alg ( L Ω Alg( or o he se of codewords C = Cod [ Alg( ] L( Ω Alg( ( m= ( m m m C( = ( C( C( C( Alg ( L Ω Alg( Defiiio 7 We defie a ecodig fucio for HG- ad HG - ivaria Reed-Solomo codes as ( ( HG-RS: Alg ( L Ω Alg( HG -RS: Alg ( L Ω Alg( i he followig forms A message m = ( m m m wih mi A lg( are rasformed by ad : C( m c( m C( m c( m C( c( = m = m C ( c ( C ( c ( Hece geeraor marices for HG- ad HG - ivaria Reed-Solomo codesare he geeralized ourier marices ad Covoluioal cyclic codes (CC s for shor form a impora class of error-correcig codes i egieerig pracice The mahemaical heory of hese codes has bee se off by hesemial papers of orey [4] ad Massey e al [5] Defiiio 8 HG- ad HG - ivaria covoluioal codes of legh ad dimesio are ideals h ( G ( L Ω lg( L Ω A lg ( havig he followig forms g of ( A ad ( (6 where ( ' ( ad C ( = ( G m ( = G ( m ( c ( = h m ( = h ( m Ω $ Ω IV Ieraioal Coferece o "Iformaio Techology ad aoechology" (ITT-8 344

8 Daa Sciece V G Labues ad E Osheimer ( ( H ( = h ( = H ( H ( H ( Alg ( ( ( g( = G ( = g( g( g( Alg ( We call marices G = G( $ ad = [ h ( ] H ' ecoders Ω Ω I is easy o see ha cyclic covoluioal codes ad group covoluioal codes are paricular cases of HG- ad HG - ivaria covoluioal codes 3 Examples Le H be a fiie Abelia group of order = The fudameal srucure heorem for fiie Abelia group implies ha we may wrie H as he direc sum of cyclic groups H = Z Z Z where Z ideified wih he rig of iegers Z l uder wih respec o l modulo l ad a eleme H is ideified wih a poi = ( of D discree orus The addiio of wo elemes H is defied as σ= = ( σ σ σ = ( H Z Z Z The ourier rasforms i he space of all fucios defied o he fiie Abelia group H = Z ad wih heir values i he fiie commuaive rig (field or some fiie algebra A has l l= a grea ieres for digial sigal processig Deoe his space as L( H A lg l ( Le ε l a primiive h roo i he algebra A lg( Le us cosruc he followig fucios ll χ ( =ε = They form he se of characers of he cyclic group l l l l l of all characers of he group where ( H ca be describe by he followig way ( ( ( Z l The hese χ =χ =ε ε ε (6 = The se of all characers { χ ( } H ad he se of all idexes H forms isomorphic muliplicaive ad addiive groups respecively wih respec o muliplicaio of characers ad addiio of idexes χ ( χ m( =χ ( =χl( where m H H l = m= ( l l l = ( m m m The followig marix [ ] Z Z Z = χ forms ourier rasform o H = ( H H The se H is called he dual group I forms frequecy domai If iiial group has he srucure H = Z Z Z he he dual group has he same srucure H = H Le us embed fiie groups H ad H io wo discree segmes Ω= [ ] ad Ω = [ ] H Ω= [ ] H Ω = [ ] (7 respecively or his aim we briefly describe a mixed radix umber sysem ow A umber sysem is called a weighed umber sysem if ay umber ca be uiquely expressed i he followig form = w i i for some se of iegers i called digis ad i w i s called weighs If he weighs are successive powers of he same umber (for example or he umber sysem is called a fixed radix umber sysem (for example radix or radix Ay umber i mixed radix + umber sysem ca be expressed i he form = i j Le where i= j= i+ + + be a fiie se of posiive iegers The wih respec o he mixed radixes above ay oegaive ieger [ ] where = ca be uiquely expressed as IV Ieraioal Coferece o "Iformaio Techology ad aoechology" (ITT-8 345

9 Daa Sciece V G Labues ad E Osheimer + i = ( = ( ( + ( + = i j i= j= + [ ] [ ] [ ] [ ] The weighs of i is j where The weigh of is uiy ( + = The radix- represeaio is = = + i i = i Le = ( H ad ( H he expressios = i j i= i= j= + + i = i j defie he maps (7 i= j= + The followig operaors (wih respec o which all characers are ivaria eigefucios T χ (: = χ ( χ ( = χ ( Ω ad are called commuaive H ( ( H T χ (: = χ ( χ ( = χ ( $ Ω ( ( H χ (: = χ( χ ( = χ ( Ω H D + i j= + (8 Dχ (: = χ( χ ( = χ $ ( Ω H (9 geeralized "ime" shif ad "frequecy" shif operaors iduced a abelia group H Iiduces exoic shifs i segmes H Ω= [ ] H oo which we will deoe as = ( Ω= [ ] H H m= m m m Ω = ( [ ] Isead of spaces L( H ( A lg ad L( H lg( ad L( Ω A lg( iervals: L( Ω A lg( H ad L( Ω A lg( H Defiiio 9 H - ad ( lg( Ω = [ ] A we will spea abouspaces L( Ω A lg( ad if ecessary i his desigaios we will disiguish groups acig i H - ivaria bloc codes L ( Ω lg( H C L Ω A H are liear bloc codes wih he propery ha if ( respecively I meas ha HypSym ( Tc ( = c ( Ω H C ad ( DC C H - ad H - ivaria { C } H ad HypSym{ C } H C A ad c Cad C( C he ( = ( Ω H C bloc codes have ordiarygroup symmeries The seclass of codes are called he abelia group codes [3 4-6] Special cases of abelia group codes are a cyclic code whe H = Z Z Z Z is a cyclic group a dyadic code whe H = Z Z Z I he firs case = = ε is he ordiary ourier rasformwhere Z Z ε a primiive h roo i he algebra lg( A ad i he secod oe ( he scalar producs ofwo vecors ( = = is he Walsh rasform where is H H = H ad ( H : = i i i= IV Ieraioal Coferece o "Iformaio Techology ad aoechology" (ITT-8 346

10 Daa Sciece V G Labues ad E Osheimer = = (+ Le = = ε ε = ε The ( - ( ( = f ( = f( ε ε f( = ( = ( ε ε Ω is direc ad iverse modulaio ourier rasform where ε is a primiive h roo i he algebra A lg( Accordig o defiiio 4 for ϕ ( = εε we have ˆ ( + ( + ( + ( + T ϕ ( =ϕ ( ( =ϕ ( ϕ ( =ε ε =ε ˆ ( + ( + ( + ( ϕ ( =ϕ (' =ϕ ( ϕ ( =ε ε =ε T ( + ( + ( + ( + ( + ( + ( + ( + ( + ( Bu ε =ε ε =ε ε = ε = ε Hece ˆ are egacyclic (sew-cyclic GSOs: ˆ T { c ( } = ( c( c ( c( ( = ( c c+ c c c c c Tˆ c ( = ( c' c ' c( ' = ( c c c c c c ( They geerae egacyclic (sew-cyclic codes Le is direc ad iverse ε - ( ( ( m+ m = m = T = = ε ε = ε The ( = f ( = f( ε ε f( = ( = ( ε ε m Ω m m ad Tˆ -modulaio ourier rasform where ε m is a primiive m h roo i he algebra A lg( Accordig o defiiio 4 for ϕ ( = εmε we have ˆ ( m+ ( m+ ( m+ ( + T ϕ ( =ϕ ( ( =ϕ ( ϕ ( =ε ε =ε Bu GSOs: m m m ˆ ( m+ ( m+ ( m+ ( ϕ ( =ϕ (' =ϕ ( ϕ ( =ε ε =ε T m m m ( + ( + ( + ( + ( + ( + ( + m m m m m m m m m m m ε =ε ε =ε ε =ε ε Hece ˆ T ad T ˆ ˆ T c ( = ( c( c ( c( ( = ( c c+ c c εmc εmc εmc Tˆ c ( = ( c' c ' c( ' = ( εmc εmc εmc c c c ( They geerae cosacyclic codes codes are cosacyclic 4 Coclusio I his paper we sudied a ew class of codes wih geeralized symmery They are ivaria wih respec o a family of geeralized shif operaors HGor HG I paricle case whe his family is a group (cyclic or Abelia hese codes are ordiary cyclic ad group codeswe deal wih GSOivaria codes wih fas code ad ecode procedures based o fas geeralized ourierrasforms 5 Refereces [] Berleamp E R 969 egacyclic codes for he Lee meric Combiaorial Mahemaics ad is Applicaios [] Berleamp E R 984 Algebraic codig heory (ew Yor: Aegea Par Press [3] Sergio R L P ad Beigo R 9 Dual geeralizaios of he cocep of cycliciy of codes Adv Mah Commu 3( IV Ieraioal Coferece o "Iformaio Techology ad aoechology" (ITT-8 347

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