1 Complemetay Dual Subfield Liea Codes Ove Fiite Fields Kiagai Booiyoma ad Somphog Jitma,1 Depatmet of Mathematics, Faculty of Sciece, Silpao Uivesity, Naho Pathom 73000, hailad e-mail : ai_b_555@hotmail.com Depatmet of Mathematics, Faculty of Sciece, Silpao Uivesity, Naho Pathom 73000, hailad e-mail : jitma_s@silpao.edu Abstact : wo families of complemetay codes ove fiite fields whee = ae studied, is suae: i) Hemitia complemetay dual liea codes, ad ii) tace Hemitia complemetay dual subfield liea codes. Necessay ad sufficiet coditios fo a liea code (esp., a subfield liea code) to be Hemitia complemetay dual (esp., tace Hemitia complemetay dual) ae detemied. Costuctios of such codes ae give togethe thei paametes. Some illustative examples ae povided as well. Keywods : Complemetay dual codes; Hemitia ie poduct; tace Hemitia ie poduct; subfield liea codes. 010 Mathematics Subject Classificatio : 94B05, 1E0. 1 Itoductio Liea codes with Euclidea Complemetay dual have bee studied i [3]. he chaacteizatio ad popeties of such codes wee give. hese codes ae iteestig sice they each the maximum decodig capability of adde chael [3]. Moeove, 1 Coespodig autho.
i some cases, such codes ca be decoded faste tha othe liea codes usig eaest eighbo decodig. I [6], ecessay ad sufficiet coditios fo cyclic codes to be Euclidea complemetay dual have bee detemied. Hemitia complemetay dual cyclic codes ove fiite fields have bee chaacteized i [5]. Subfield liea codes ad thei duals ude the tace Hemitia ie poduct have bee studied i [1] ad [4]. Such codes have a applicatio i costuctig uatum codes i [1] ad efeeces theei. o the best of ou owledge, Hemitia complemetay dual liea codes ad tace Hemitia complemetay dual subfield liea codes have ot bee well studied. heefoe, it is of atual iteest to studied complemetay dual codes with espect to the Hemitia ad tace Hemitia ie poducts. I this pape, we focus o Hemitia complemetay dual liea codes ad tace Hemitia complemetay dual subfield liea codes. Chaacteizatios, popeties, ad costuctios of such codes ae studied. he pape is ogaized as follows: Some basic cocepts ad pelimiay esults o complemetay dual codes ae ecalled i Sectio. I Sectio 3, chaacteizatio of complemetay dual codes with espect to the two ie poducts ae give. Some costuctios ad illustative examples of such complemetay dual codes ae established i Sectio 4. Pelimiaies Let ad = be pime powe iteges ad let be fiite fields. Let : deote the tace map give by ( ) = fo all. Some popeties of the tace map ca be foud i [, heoem.3]. Fo u = ( u1, u,, u), let 1 u = ( u, u,, u ), whee a = a fo all a. Fo each matix A [ aij ] Mm, ( ), let A= [ a ij ] ad ( A) = [( a ij )]. Give uv,, let w t( v ) deote the Hammig weight of v ad let d( uv, ) deote the Hammig distace betwee u ad v.
3 A code of legth ove is defied to be a oempty subset C of. he miimum distace dc ( ) is give by d( C) = mi{ d( u, v) u, v C, u v }. A [,, d ] liea code C is a -dimesioal -subspace of with miimum distace d. A matix G ove a [,, d ] liea code C if the ows of G fom a basis of C. Fo a geeal code is called a geeato matix fo C, the otatio (, M = C, d ) is commoly used. A code C is said to be a -liea code ove if C is a subspace of the - vecto space. Whe is clea fom the cotext, C is called a subfield liea code ove. It is ot difficult to see that if C is a -liea code of legth ove, the C = fo some 0 ad dim ( C ) =. A matix G ove is called a geeato matix fo a (,, d ) -liea code C if the ows of G fom a basis of C as a -vecto space. Fo u = ( u1, u,, u ) ad v = ( v1, v,, v ) i betwee u ad v ae defied as follows:, the ie poducts uv, := i uv i i is the Euclidea ie poduct of u ad v. (1) E =1 () u, vh:= i=1uv i i = u, v E is the Hemitia ie poduct of u ad v. (3) he tace Hemitia ie poduct ae defied ito two cases depedig o the field chaacteistic: that = (a) Fo a eve, u, vh := ( u, v H). (b) Fo a odd, u, vh := ( u, v H), whee \{0} is such. he Euclidea dual (esp., Hemitia dual ad tace Hemitia dual) of a code C is defied to be the set
4 C E := { u u, c 0 fo all c C} (esp., H C := { u u, ch 0 fo all c C} ad H C := { u u, ch 0 fo all c C}). A code C of legth ove E is said to be Euclidea (esp., Hemitia ad tace Hemitia) complemetay dual if C E = { 0 } (esp., C H = { 0 } ad CC H = { 0 } ). Next popositio follows fom the defiitios of complemetay dual codes. Popositio.1. Let C be a code of legth ove =. he the followig statemets hold. i) If C is a liea code, the C is Euclidea complemetay dual if ad oly if C = E. C C ii) If C is a liea code, the C is Hemitia complemetay dual if ad oly if = H. C C C iii) If C is a ad oly if -liea code, the C is tace Hemitia complemetay dual if = C C H. he followig popeties of codes ad thei duals ae discussed i [4, Chapte 3]. Popositio.. Let C be a code of legth ove =. he the followig statemets hold. E = i) If C is a liea code, the C E C ad H C H = C. H = ii) If C is a -liea code, the C H C.
5 Note that the popeties C tue if C is ot a liea code. H H = C ad C E E = C do ot eed to be he followig popeties ae diect coseuece of Popositio.. Coollay.3. Let C be a code of legth ove =. he the followig statemets hold. i) If C is a liea code, the ( ) ( E dim C dim C ) ad dim C C. ( ) ( H dim ) ii) If C is a -liea code, the = ( ) ( H dim C dim C ). Fom Coollay.3, to study complemetay duality of codes, we focus o the Euclidea ad Hemitia ie poducts if codes ae liea, ad the tace Hemitia ie poduct if codes ae -liea ove. 3 Chaacteizatio of Complemetay Dual Subfield Liea Codes he chaacteizatio ad popeties of Liea codes with Euclidea Complemetay dual have bee established i [3]. I this sectio, chaacteizatios of Hemitia complemetay dual liea codes ad tace Hemitia complemetay dual subfield liea codes ae give i tems of othogoal pojectios. Defiitio 3.1. Let V be a ie poduct space ove a field. A -liea map : V V is called a -othogoal pojectio with espect to the pescibed ie poduct, if i) =, ad ii) uv, = 0 fo all u Im( ) ad v e( ).
6 3.1 Chaacteizatio of Hemitia Complemetay Dual Liea Codes he followig popety of -othogoal pojectio plays vital ole i chaacteizig Hemitia complemetay dual liea codes ove Lemma 3.. Let C be a liea code of legth ove = ad let : be a -liea map. he is a -othogoal pojectio with espect to the Hemitia ie poduct oto C if ad oly if Poof. Suppose that : Hemitia ie poduct oto C. Let v if vc, ( v) = 0 v if C H. C is a -othogoal pojectio with espect to the v C ad u C H have C Im( ). Hece, thee exists a wod v ( x) ( x) ( ( x)) ( v ). Sice u C H, we have v C Im( ). It follows that u e( ), ad hece, ( u ) 0. Covesely, assume that v if vc, ( v) = 0 v Sice is a fuctio, C H = { 0 }. Fo each C. Sice is oto C, we x such that ( x) = v ad if C H. v uv, 0 fo all H, it ca be witte uiuely as v = u w, whee v C ad w C H. he ( u) = u ad ( w) = 0. Hece, ( u) = ( ( u)) = ( u ) ad It follows that ( v) = ( v ) fo all ( w) = ( ( w)) = ( 0) = 0 = ( w ). v. Let u Im( ) ad v e( ). he u C ad ( v) = 0. It follows that v C H ad uv, H = 0. Hece, Im() ad e() ae othogoal with espect to the Hemitia ie poduct. follows. he chaacteizatio of Hemitia complemetay dual liea codes is give as
7 Lemma 3.3. Let C be a liea code of legth ove =. he C is Hemitia complemetay dual if ad oly if thee exists a with espect to the Hemitia ie poduct fom Poof. Assume that ie poduct fom oto C. -othogoal pojectio C is a -othogoal pojectio with espect to the Hemitia oto C. By Lemma 3., we have C v if vc, ( v) = 0 v if C H. Suppose that C is ot Hemitia complemetay dual. he the exists that u 0 such u CC H, i.e., u C ad u C H. It follows that 0 u = C ( u) = 0, a cotadictio. heefoe, C is Hemitia complemetay dual. Covesely, suppose C is Hemitia complemetay dual. Let thee exists a uiue pai C : by It is ot difficult to veify that heefoe, by Lemma 3., v. he u C ad w C H such that v = u w. Defied a map C C ( v ) = u. C is a liea map such that z if zc, ( z) = 0 z if C H. C a -othogoal pojectio with espect to the Hemitia ie poduct fom oto C. heoem 3.4. Let C be a, liea code ove = with geeato matix G. he C is Hemitia complemetay dual if ad oly if I this case, 1 := ( ) is a C G GG G to the Hemitia ie poduct fom oto C. GG is ivetible. -othogoal pojectio with espect
8 Poof. Suppose that we have a( GG ) < GG is a o-ivetible matix. Sice. It follows that = ull( GG ) a( GG ) < ull( GG ). GG is a matix, he ull( GG ) > = 0, i.e., { 0 } e( GG ). he thee exists ue( GG ) \{ 0 }. Hece, ugg = 0 ad ugc\{ 0 }. Fo each v C, it ca be witte as v= u G fo some u. Hece, heefoe, ug, v = ( ug) v = ( ug)( u G) = ugg ( u) = 0( u) = 0. H ug 0 is also a wod i C H. It follows that C H {} 0, i.e., C is ot Hemitia complemetay dual. exists Covesely, assume that GG is ivetible. Let u such that v= u G, ad hece, C v. If v C, the thee 1 1 vg ( GG ) G = ugg ( GG ) G = ui G = ug = v. If v, the vg = 0, ad hece, C H 1 1 vg ( GG ) G = 0( GG ) G = 0. heefoe, 1 G ( GG ) G is a Hemitia ie poduct fom complemetay dual by Lemma 3.3. -othogoal pojectio with espect to the oto C. heefoe, C is Hemitia Example 3.5. Let C be a liea code of legth 4 ove 4 = {0,1,, = 1} 1 0 0 with geeato matix G = 0 1 1. Sice 0 GG =, we have 1
9 det( GG ) = 1 which implies that complemetay dual by heoem 3.4. GG is ivetible. Hece, C is Hemitia 3. Chaacteizatio of Complemetay Dual Subfield Liea Codes. Now, we focus o the chaacteizatio of tace Hemitia complemetay dual subfield liea codes. Lemma 3.6. Let C be a -liea code of legth ove = ad let : be a -liea map. he is a -othogoal pojectio with espect to the tace Hemitia ie poduct oto C if ad oly if v if vc, ( v) = 0 v if C H. Poof. Usig the agumets simila to those i Lemma 3. ad applyig the tace Hemitia ie poduct istead of the Hemitia ie poduct, the statemet is poved. Lemma 3.7. Let C be a -liea code of legth ove =. he C is tace Hemitia complemetay dual if ad oly if thee exists a pojectio with espect to the tace Hemitia ie poduct fom Poof. Assume that C is a Hemitia ie poduct fom C -othogoal oto C. -othogoal pojectio with espect to the tace oto C. By Lemma 3.6, it follows that v if vc, ( v) = 0 v if C H. Suppose that C is ot tace Hemitia complemetay dual. he thee exists u 0 such that u C H. It follows that 0 u = C ( u) = 0, a cotadictio. C Hece, C is tace Hemitia complemetay dual.
10 Covesely, suppose that C is tace Hemitia complemetay dual. Let v. he thee exists a uiue pai u C ad w C H such that C v = u w. Defied a map : by C ( v ) = u. It is ot difficult to see that C is a -liea map such that Hece, by Lemma 3., C C a Hemitia ie poduct fom z if zc, ( z) = 0 z if C H. -othogoal pojectio with espect to the tace oto C. heoem 3.8. Let C be a (, ) -liea code ove = with geeato matix G. he C is tace Hemitia complemetay dual if ad oly if GG GG is ivetible. C C I this case, : defied by is a 1 ( vg )( GG GG ) G if is eve, C ( v) = 1 1 ( vg )( GG GG ) G if is odd -othogoal pojectio with espect to the tace Hemitia ie poduct fom oto C, whee \{0} is such that =. Poof. Assume that cases. GG GG is ot ivetible. We distiguish the poof ito two Case 1 is eve. he ( GG ) = GG GG is ot ivetible. Sice ( GG ) is a matix, we have a(( GG )) <. It follows that = ull(( GG )) a(( GG )) < ull(( GG )).
11 Hece, ull(( GG )) > = 0, i.e., { 0 } e(( GG )). he thee exists ue(( GG )) \{ 0 } such that u(( GG )) = 0 ad ugc\{ 0 }. Hece, ( ugg ) = ( ug) G ugg = u(( GG )) = 0. Case is odd. he ( GG ) = ( GG GG ) is ot ivetible fo all \{ 0 } such that =. Sice ( GG ) is a matix, we have a(( GG )) < ad = ull(( GG )) a(( GG )) < ull(( GG )). It follows that ull(( GG )) > = 0, ad hece, { 0 } e(( GG )). he thee exists ue(( GG )) \{ 0 } such that u(( GG )) = 0 ad ugc\{ 0 }. We have ( ugg ) = (( ug) G ugg ) = u(( GG )) = 0. Altogethe, it follows that u G is also a wod i C H, ad hece, CC H {} 0. heefoe, C is ot tace Hemitia complemetay dual. Covesely, assume that det( GG GG ). Let : be defied by C C 1 ( vg )( GG GG ) G if is eve, C ( v) = 1 1 ( vg )( GG GG ) G if is odd. Let v. If v C, the thee exists u such that v= u G, ad hece, 1 ( vg )( GG GG ) G if is eve, C ( v) = 1 1 ( vg )( GG GG ) G if is odd, 1 ( ugg )( GG GG ) G if is eve, = 1 1 ( ugg )( GG GG ) G if is odd,
1 Assume that ad 1 ( ugg ugg )( GG GG ) G if is eve, C ( v) = 1 1 ( ugg ugg )( GG GG ) G if is odd, = u( GG GG )( GG GG ) G = ui G = ug = v. v C H. he 1 ( vg ) if is eve, 0 = ( vg ) if is odd Hece, C is a poduct fom 1 ( vg )( GG GG ) G if is eve, C ( v) = 1 1 ( vg )( GG GG ) G if is odd, 1 0( GG GG ) G if is eve, = 1 1 0( GG GG ) G if is odd, = 0. -othogoal pojectio with espect to the tace Hemitia ie oto C. heefoe, C is tace Hemitia complemetay dual. Example 3.9. Let 9 = 3( ), whee is a oot of liea code of legth 4 ove Sice 9 x with geeato matix x. Let C be a 3-3 1 0 0 0 1 G =. 4 0 0 3 0 GG 5 7 0 0 1 0 0 1 0 0 0 7 5 6 6 6 0 0 0 0 0 0 GG =, 6 3 6 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0
13 we have that GG GG is ivetible. Hece, by heoem 3.8, C is tace Hemitia complemetay dual. Example 3.10. Let 4 = {0,1,, = 1} ad let C be a -liea code of 1 0 0 0 1 1 legth 4 ove 4 with geeato matix G =. Sice 0 0 0 GG 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 GG =, 0 0 0 0 0 1 0 1 1 1 1 1 0 1 1 0 it follow that GG Hemitia complemetay dual. GG is ivetible. heefoe, by heoem 3.8, C is tace 4 Costuctios of Complemetay Dual Subfield Liea Codes I this sectio, some costuctios of complemetay dual codes with espect to the Hemitia ad tace Hemitia ie poducts ae give. 4.1 Costuctios of Hemitia Complemetay Dual Liea Codes It is well ow that, fo a give [,, d ] code, thee exists a euivalet code with the same paametes such that its geeato matix is of the fom G = [ I A ] fo some ( ) matix A ove. he geeato matix of a liea code of this fom plays a impotat ole i costuctig Hemitia complemetay dual codes. he followig fact is well ow. Lemma 4.1. Let p be a pime. he 1 is a uadatic modulo p if p 1 mod 4.
14 Some costuctios of Hemitia complemetay dual subfield liea codes ae give as follows. heoem 4.. Let C be a [,, d ] liea code of legth ove = geeato matix G = [ I P ]. he the followig statemets hold. with i) If cha( ) =, the a liea code C geeated by matix G = [ I P P ] is Hemitia complemetay dual with paametes [,, d ], whee ii) If cha( ) 1 mod 4, the thee exists such that liea code d d. = 1 ad a C geeated by G = [ I P P] is a [,, d ] Hemitia complemetay dual code, whee d d. Poof. i) Assume that cha( ) =. he heefoe, G( G) = I PP PP = I PP = I 0 = I. GG is ivetible. he code C geeated by G is Hemitia complemetay dual by heoem 3.4. Sice C is a liea code of legth, G has colums. Note that P has colums. It follows that G = [ I P P] has ( ) ( ) = colums. Hece, C geeated by G is a [, ] liea code. Next, we show that d( C) d. Let vc \{ 0 }. he thee exists u \{ 0 } such that v = ug = [ ui up u P]. Hece, heefoe, d= d( C) d. wt( v) = wt([ ui up up]) wt([ ui up]) = wt( u[ I P]) = wt( ug) = d( ug) d( C) = d. ii) Assume that cha( ) 1 mod 4. he = 4 1 fo some itege. By Lemma 4.1, thee exists such that = 1. he
15 heefoe, 1 G( G) = I PP PP 4 1 1 = I PP PP = I PP ( 1) PP = I. GG is ivetible. Hece, by heoem 3.4, C geeated by G is Hemitia complemetay dual. Simila to i), we ca pove that C is a[,, d] code with d d. Example 4.3. Let C be a liea code of legth 4 ove with the geeato matix heoem 4., a liea code geeated by complemetay dual with paametes [6,,] 4. Example 4.4. Let 5 = 5( ), whee is a oot of 4 = {0,1,, = 1} 1 0 0 G = 0 1 1. he C is a [4,,] 4 code. By 1 0 0 0 G = 0 1 1 1 is Hemitia x 4x. Let C be a liea 5 code of legth 4 ove 5 with the geeato matix 1 0 G =. he 0 1 19 C is a [4,,3] 5 code. Sice geeated by 1mod 5, by heoem 4., the code C 5 5 1 0 G = 0 1 19 19 is Hemitia complemetay dual with paametes [6,, dc ( ) 3] 5. 4. Costuctio of Complemetay Dual Subfield Liea Codes Give a (,, d ) -liea code C ove = ( ), a geeato matix of C is a = matix ove. Usig elemetay ow ad colum
16 opeatios, thee exists a euivalet geeato matix -liea code with the same paametes with I A G = I A 0 B fo some itege 0, ( ) matix A ove, ad ( ) ( ) matix B ove, whee 0 deotes the ( ) matix whose eties ae 0. Some costuctios of tace Hemitia complemetay dual codes ae give as follows. heoem 4.5. Let C be a (,, d ) -liea code C ove geeato matix = = ( ) with I A G = I A 0 B such that BB BB is ivetible. he the followig statemets hold. i) If cha( ) =, the a -liea code C geeated by I A A 0 G = I A A 0 0 B B B is tace Hemitia complemetay dual with paametes (3,, d ), whee d d. ii) If cha( ) 1 mod 4, the thee exists such that -liea code C geeated by I A A 0 G = I A A 0 0 B B B = 1 ad a
17 is tace Hemitia complemetay dual with paametes (3,, d ), whee d d. Poof. I both cases, the paametes ca be veified usig agumets simila to those i heoem 4.. Next, we show that i) Assume that cha( ) =. he C is tace Hemitia complemetay dual. ad Sice BB I I 0 1 GG = I I 0 0 0 BB 0 ( ) I 0 GG GG = ( ) I 0 0. 0 0 BB BB BB is ivetible, GG GG is osigula. By heoem 3.8, the -liea code C geeated by G is tace Hemitia complemetay dual. ii) Assume that cha( ) 1 mod 4. he = 4 1 fo some. By Lemma 4.1, thee exists such that = 1. he 1 (1) = = 1, ad hece, 1 1 1 I (1 ) AA ( I (1 ) AA ) (1 ) AB 1 1 1 1 = ( (1 ) ) ( (1 ) ) (1 ) 1 1 1 GG I AA I AA AB (1 ) BA (1 ) BA (1 ) BB BB I I 0 1 = I I 0. 0 0 BB Coseuetly, we have
18 which is ivetible if ad oly if 0 ( ) I 0 GG GG = ( ) I 0 0 0 0 BB BB BB BB is ivetible. heefoe, the -liea code C geeated by G is tace Hemitia complemetay dual by heoem 3.8. Example 4.6. Let legth 4 ove 4 4 ad let C be a -liea code of = {0,1,, = 1} with the geeato matix 1 0 0 0 1 0 0 0 G =. 0 0 0 0 1 0 0 1 he C is a 6 (4,,) -liea code. 4 the 1 1 1 1 0 Sice 1 1 1 1 0 -liea code C geeated by G 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 = 0 0 0 0 0 is ivetible, is Hemitia complemetay dual with paametes 4.. By diect calculatio, we have dc ( ) = 3. 6 (8,, dc ( ) ) 4 by heoem
19 Acowledgemets his eseach is suppoted by the hailad Reseach Fud ude Reseach Gat RG5780065. Refeeces [1] M. F. Ezema, S. Jitma, S. Lig, D. V. Pasechi. CSS-lie costuctios of asymmetic uatum codes. IEEE as. If. heoy, 59 (013) 673 6754. [] R. Lidl, H. Niedeeite, Fiite Fields. Ecyclopedia of Mathematics ad its Applicatios. vol. 0. Cambidge: Cambidge Uiv. Pess; 1997. [3] J. L. Massey. Liea codes with complemetay duals. Discete Mathematics, 106/107 (199) 337 34. [4] G. Nebe, E. M. Rais, ad N. J. A. Sloae. Self-Dual Codes ad Ivaiat heoy. Beli:Spige; 006. [5] E. Sagwisut, S. Jitma, S. Lig, P. Udomavaich. Hulls of cyclic ad egacyclic codes ove fiite fields. Fiite Fields ad hei Applicatios, 33 (015) 3 57. [6] X. Yag, J. L. Massey. he coditio fo a cyclic code to have a complemetay dual. Discete Mathematics, 16 (1994) 391 393. (Received 11 Octobe 015) (Accepted 31 Jauay 016)