Equivalence of Codes over Finite Chain Ring. Sonlu Zincir Halkası üzerinde Kodların Denkliği

Transcription

1 Joral of New Reslts i Egieerig ad Natral Sciece, No:8 (08) Received: November 6, 07 Accepted: Jaary 6, 08 Eqivalece of Codes over Fiite Chai Rig stafa ÖZKAN * Trakya Uiversity, Faclty of Scieces, athematics Departmet, 030, Edire, Trkey Abstract Eqivalece of Codes are costrcted classificatio o fiite rigs accordig to odd codes, eve codes ad recrret codes It is determied that the codes are odd codes or eve codes ad the relatioship with the Hadamard codes was stdied I this stdy, some matrices is writte with the elemets o the fiite rigs Codes are geerated with these matrices The relatioship betwee the geerated codes ad the perfect codes is revealed Keywords: Odd codes, Eve codes, Gray map, Hadamard codes, Special matrices Sol Zicir Halkası üzeride Kodları Dekliği stafa ÖZKAN * Trakya Uiversity, Faclty of Scieces, athematics Departmet, 030, Edire, Trkey Özet Tek kodlar, çift kodlar ve tekrarlı olşm kodlarıa göre sol halkalarda kodları dekliğii sııfladırılması krlmştr Kodları tek kod ya da çift kod oldğ ve çalışıla Hadamard kodları ile ilişkisi tespit edilmiştir B çalışmada, bileşeleri sol halkalarda matrisler yazılmıştır Kodlar b matrisler ile üretilmiştir Üretile kodlar ve mükemmel kodlar arasıda ilişki ortaya kolmştr Aahtar Kelimeler : Tek kodlar, Çift kodlar, Gray döüşümü, Hadamard kodlar, Özel matrisler * Correspodig Athor, e- mail: mstafaozka@trakyaedtr, mstafaozka@iclodcom 48

2 Joral of New Reslts i Egieerig ad Natral Sciece, No:8 (08) Itrodctio Biary operatios (+ ad ) o the rig,,,,,, R IF IF IF defied as below: IF IF IF 8) are (

3 The Lee weight w L (r) of r R is give by Joral of New Reslts i Egieerig ad Natral Sciece, No:8 (08) ; r 0 w L ( r) 4 ; r ; otherwise () This exteds to Lee weight fctio i R sch that w L (r) = i0 w ( ) for L r i r ( ro, r,, r ) R The Lee distace d L ( x, y) betwee ay distict vectors, is defied to be w L ( x y) The x y R d L miimm Lee distace of C is defied as d L ( C) mi{ d L ( x, y)} for ay The Hammig weight of C is defied as where if ad if The miimm Hammig distace of C is called as d ( C) mi d ( c, c) H for ay H Geerally the Gray map is defied as : : R IF 4 ( r, r,, r ) a ( r, r,, r ) ( c, c,, c, a c, a c,, a c, b c, b c,, b c, a b c, a b c,, a b c ) () where r i ai bi ci R for i aterials ad ethods Certai examples for the matrix, costrcted above are give below : x 0,0,,0 0, x ,, 0 This matrix costrcted is a special matrix which has rows x8 3x4 50

4 Joral of New Reslts i Egieerig ad Natral Sciece, No:8 (08) Defie the code C { ( c, c ) c R, c IF } which has a geerator matrix, where, are iregers sch that, 0, Defiitio : Let C R,, be a code ( ) { (,,, ) eve C c c c R ( c, c,, c ) C } 3 3,, is called a eve code over R ( ) { (,,, ) odd C c c c R ( c, c,, c ) C } is called a 4 4 odd code over R Eve codes ad odd codes are defied over the field F writig F istead of R 3 Reslts, Prepositio 3 : Let C be a code,,, i) eve( C ) odd( C ) C is satisfied if, 0,,, ii) eve( C ) odd( C ) C is satisfied if 0,,, Prepositio 3 : eve( ( C )) ( eve( C )) is satisfied, Proof : Let r ( r, r,, r ) C where ri ai bi ci R for i If ( r) ( r, r,, r ) ( a b c, a b c,, a b c ) ( c, c,, c, a c, a c,, a c, b c, b c,,, b c, a b c, a b c,, a b c ) ( C ) ( c, c3,, c, a c, a3 c3,, a c, b c, b3 c3,, b c, the, a b c, a b c,, a b c ) eve( ( C )) O the other had, r, ( r, r,, r ) ( a b c, a b c,, a b c ) eve( C ) The ( c, c3,, c, a c, a3 c3,, a c, b c, b3 c3,, b c, ( r), a b c, a b c,, a b c ) ( eve( C )) This Prepositio ca be writte for the odd codes 5

5 Joral of New Reslts i Egieerig ad Natral Sciece, No:8 (08) Example 33 : Write the matrix x to defie the code C 0 The the elemets of the code C are of the form c ( c, c ),where c R, c IF C { 0 0,0,,,,,, 0,,,,,,,, } R It is see that d L ( C ) 4 ad C 6 ad the this is a (,6,4) _ codetherefore is a ( C ) { , 0000, 0000, 000, 0000, , 0000, 0000, 0000, 0000, ,0000, 0000} IF 8 ( 8,6,4) _ Hadamard code Let A 4 4x4 be a ormalized Hadamard matrix Writig 0 istead of ad istead of, the vectors 0000,000 ad 00 are obtaied Addig the complemets of these vectors to these vectors the ew vectors 0000,000, 00,,00, 00 ad 00 are obtaied The sig the method give above, the ew codewords , 0000, 0000, 0000,, , 0000,0000,0000,0000, 0000,0000,0000, , are obtaied The code formed by these codewords is ( C ) which is a ( 8,6,4) _ Hadamard code oreover ( C ) 00,,, IF ad (( C ) ) { , 0000,, 0000} IF 8 5

6 Joral of New Reslts i Egieerig ad Natral Sciece, No:8 (08) C is a cyclic code sch that the eqatio 0 ( C ) C is provided Similarly ( C ) is qasi- 4 cyclic code of idex 4 sch that the eqatio ( ( C )) ( C ) is satisfied For the code C ; eve( C ),,,,,, C 0,0 R is obtaied 0000, 00,,00, ( eve( C )) 00, 0000 Hece 0, 0, eve( ( C )) ( eve( C )) is obtaied 4 Coclsios I this paper the codes over the a fiite chai IF IF IF where 3 0 defiig by matrices give i sectio are Classified The codes over the rig defiig by matrices give lexicographically featre are described These codes over the rig are obtaied images of codes i the Galois field oreover eve codes, odd codes ad recrret formsa are obtaied Ackowledgemets The athor is gratefl to the aoymos referee for a carefl checkig of the details ad ISSIT07 5 Refereces [] Qia J, Zhag L, Zh S (006), "Costacyclic ad cyclic codes over F F F ",IEICE TrasFdametals, E89-a, o 6, [] Özka, Öke F(06), Some Special Codes Over F3 vf3 F3 F, athematical Scieces 3 ad Applicatios E-Notes Vol 4 No,pp [3] Roma S (99), Codig ad Iformatio Theory, Gradate Texts i athematics, Spriger Verlag m [4] Udomkavaich P, Jitma S (009), O the Gray Image of ( )-Cyclic Codes m F k F k F, ItJCotemp ath Scieces,Vol4, No6, 65-7 k p p p [5] Özka, Öke F(06),A relatio betwee Hadamard codes ad some special codes over F+F, Appathematics ad If Sci Vol0, No:, pp : [6] Özka, Öke F (07),Repeat codes, Eve codes, Odd codes ad Their eqivalece, Geeral Letters i athematics, Vol, No :, pp : 0-8 [7] Özka, Öke F (07),Codes defied via especial matrices over the rig ad Hadamard codes, athematical Scieces ad Applicatios E-Notes, Volme 5, No :, pp : [8] Özka, Öke F (07), Gray images of ( v) costacyclic codes over a particlar rig, Palestie Joral of athematics Vol 6(SI), 4-45 [9] Krotov, D S(000), Z4-liear perfect codes,diskret Aal Issled Oper SerVol 7, 4, [0] Krotov, D S(00), Z4-liear Hadamard ad exteded perfect codes, Procs of the Iteratioal Workshop o Codig ad Cryptography,Paris,