Images of Linear Block Codes over Fq ufq vfq uvfq

Transcription

1 Oe Joua of ed Sceces, 03, 3, 7-3 do:036/oas Pubshed Oe 03 (htt://wwwscog/oua/oas) Iages of Lea oc Codes ove u v uv Jae D Paaco, Vgo P Sso Isttute of Matheatca Sceces ad Physcs, Uvesty of the Phes Los años, Coege, Lagua, Phes Ea: daaco@ubeduh, vsso@ubeduh eceved 03 STCT I ths ae, we cosdeed ea boc codes ove st we ooed at the stuctue of the g It was show that u v uv u v uv vu whee,, 0, s ethe a fte cha g o a ca dea g but s a oca g We the estabshed a geeato at fo the ea boc codes ad eued t wth a hoogeeous weght fucto ed codes wee the costucted as ages of these codes by usg a bass of ove ouds o the u ag dstace of the age codes wee the deved code eetg such bouds s gve as a eae Keywods: -ay Iages; Dstace ouds Itoducto Let be a e ube,, ad deote the Gaos fed wth eeets Dug the ate 990s, C achoc used ea boc codes ove u, u 0 fo costuctg odua attces Its success otvated the study of ea boc codes ove the fte cha g u d ay of the esuts fo these studes have bee eteded ove fte cha gs of the fo u u u, u 0, Such gs ca be see as atua etesos of u othe g eteso of u s u v uv whee u v 0, uv vu Ue u, s ethe a fte cha g o a ca dea g Ydz ad S Kaadez toduced ea boc codes ove the g u v uv [6] Sef-dua codes, cycc codes ad costacycc codes ove ths g wee aso studed by these authos [3,7,8] I 0, X Xu ad X Lu studed the stuctue of cycc codes ove [5] I ths wo, we w aayze ea boc codes ove The stuctue of the g w be dscussed Secto The geeato at of ea boc codes ove ad weght fuctos defed o w be taced Secto 3 The -ay ages of these ea boc codes ad bouds o ts u ag dstace w be eseted Sectos ad 5, esectvey Lasty, a code eetg these bouds s gve Secto 6 Peaes ad Deftos Stuctue of the g u v uv Let deote the g u v uv w hose eeets ca be uuey wtte as abucv duv whee abcd,,, eeet of s a ut f ad oy f a 0 The g has 5 deas aey 0, uv, v, u, v,, u v whee s ot a ca dea g sce the aa dea uv, s geeated by u ad v The cadaty of the deas ae uv, 3 v u v, uv,, ad Its attce of deas s show gu e s ca be see the attce of deas, s ot a fte cha g ut t s a oca, Noethea ad ta g zeo dvsos ae the eeets of uv, \ 0 ad ts uts ae the eeets of \ u, v gue Lattce of deas of u v uv Coyght 03 Sces OJS

2 8 J D PLCIO, V P SISON Ceay, the g s soohc to [, y ], y, y y It s aso soohc to the g of a atces of the fo a b c d 0 a 0 c 0 0 a b a Moeove, s obeus wth geeatg chaacte t d : T, a bu cv duv e whee t deotes the tace a o ad T s the utcatve gou of ut coe ubes uthe, s a vecto sace ove wth deso bass of ove s gve by the set, uvuv,, whch we w efe to as the oyoa bass of othe bass cosdeed ths wo s u v uv, v uv, u uv,u v 3 Lea oc Codes ove uv uv y ea boc code ove a fte coutatve g has a geeato at whch ca be ut the foowg fo ai,,3, ai a,3 a, G () ai a, whee ae bay atces fo ad ae atces, ove fo code of ths fo has a eeets, whee the a ' s defe th e ozeo euvaece casses a, a,, a ude the euvae ce eato o defed by a bf a bu fo a ut u a afo soe ; ad the bas G ae to be fed wth zeos ea boc code of egth ove s a -subodue of has a geeato at whch ca be ut the fo show gue whee, ae atces ove, D, ae atces ove ad the ba ats of G ae to be 3 fed wth zeos Moeove, has t codewods whee t ea boc code ove s fee f ad oy f 0 fo a,, 3 Now, w e eu wth two weght fuctos aey the usua ag etc ad a hoogeeous we ght fucto Lea (T ood, []) Let be a obeus g wth geeatg chaacte, the evey hoogeeous weght w ho o c a be eessed tes of as foows w ho y () y whee s the gou of uts of Theoe hoogeeous weght w ho o s gve by f \ uv f uv 0 othewse \ 0 Poof: Let abucvduv Now, usg the evous ea, evey hoogeeous weght o ca be eessed as w ho 3 y y Case Let Thee ae uts havg the sae d, fo ay d ut thee ae eeets of that has tace, fo ay ece, () I,,3,,5, vi vd,3 vd, vd,5 vd, ui ud ud 3 3, ud3,5 3, ( uv) I uvd uvd G ( u v) I u vd u vd uvi uvd 3 3,,5,,, gue Geeato Mat of Lea oc Codes ove u v uv Coyght 03 Sces OJS

3 J D PLCIO, V P SISON 9 ut y e y e 0 So, Case Let uv \ 0 o evey a, thee 3 ae uts of the fo y a bu cv duv Now, of these have the sae tace vaue, fo ay whe thee ae of the wth tace zeo ece, ut y 3 3 y e e So, Case 3 Let uv, \ uv Thee ae eeets of uv, \ uv that have the sae coeffcet fo uv o each eeet uv, \ uv aeas coes the utset y y, uv, \ uv Moeove, thee ae eeets of that has tace, fo ay ece y y e We eted ths to atuay: f,,, the The hoogeeous (es ag) dstace betwee ay dstct vectos, y, deoted by dho, y (es d, y ), s defed as y (es w y ) We w deote the u hoogeeous dstace ( es ag) dstace by a ea boc code ove by (es d ho d ) The -ay Iages of Lea oc Codes ove u v uv Let b, b, b3, b be dstct eeets of a odeed bass of The ay eeet of ca be wtte the fo ab, a Defe the ag : ab a, a, a, a 3 We the eted to coodate-w se: f,,, ad a, b the a,,,,,,,,,, a, a, a, a, a, It s easy to show that s a -odue soohs Theoe If s a ea boc code ove of egth, the s a ea boc code ove wth egth Poof: st we show that fo evey, Let,,, Sce fo ay,,,, the Net we show that s a subsace of Let s ad et y, y The thee est, such that y ad y ut sy y s sce s a odue hooohs Sce s, sy y Thus, s a subsace of Theoe Let G be the geeato at of gve gue The G has a geeato a- t that s eutato-euvaet to the at gve gue 3 I,,3, vi v, v,3 v, ui u, u,3 u, uvi uv, uv,3 uv, vi vd vd,3, uvi uvd,3 uvd, u vi u vd u vd,, uvi uvd,, uvd gue 3 Geeato Mat of uvi uvd 3 3, Coyght 03 Sces OJS

4 30 J D PLCIO, V P SISON Poof: Let have a geeato at gve gue The fo evey c, c ca be eessed as yg whee y,, that s, c sz whee s ad the ' ae the ows of ay ba ss of, c ca futhe be wtte Now, ece, z s G Usg a z b uz c vz d uvz,,,,, ca, zb, uz c vz d uvz,,,,, S z uz vz uvz,,, ut vz 0 wheeve,, o 3,, ; uz 0 wheeve,, o sas 3 3,, ; uvz 0 wheeve ; ad uz vz fo soe,, fo soe wheeve Defe the set as the esutg set oce the udesabe cases sted above ae deducted fo the set S Notce that the eeets of ae the ows of the at gve gue 3 we w deote by M Now, defe as the at that cossts of the ows,, wtte the fo 3 of M so that M ca be W e wsh to show that the ows of M ae eay deedet Wthout oss of geeaty, et fo a Cosde a ow of Ceay, t caot be eessed as a ea co bato of ows fo ay of the ' s, We ow that, u, v, uv ae eay deedet ad so ay ozeo ea cobato of these vectos s ot the zeo vecto Thus, ay ow of caot be wtte as a ea cobato of ows of ay of the ' s, ece, the ows of M ae ey deedet a The succeedg theoes ae dect coseueces of Theoe Cooay 3 If s fee wth a, the s fee wth a Cooay Let be a fee ate- ea boc code ove wth geeato at I, the the geeato at of the -ay age of wth esect to the bass uvuv, vuv,uuv, uv s eutato-euvaet to 0 I I I E D E D D I 0 I 0 D E 0 D E I I 0 0 D D 0 I 0 0 I D 0 0 D whee DEuv uv 5 Dstace ouds of the Iages of Lea oc Codes ove u v uv The u dstace of a code gves a se dcato of the goodess of a code fed code ca coect at ost eos whee s ts u ag dstace ece, we ae teested wth ue bouds of the u ag dstace of the ages of the ea boc codes ove o the succeedg dscussos, we et be a ate- ea boc code ove so, we deote by the u ag dstace of Theoe 5 (Sgeto-tye oud) Let be fee The (3) The above theoe s a dect coseuece of Cooay 3 ad the Sgeto oud fo codes ove feds whe the et theoe s a dect coseuece of the Pot oud fo codes ove feds Theoe 5 (Pot-tye oud) Let be fee The () The et boud s tes of the aveage hoogeeous weght o ad the u ag ds- tace of Theoe 53 (as-tye boud) o a code, d d () Poof: Note that s bouded above by If fo evey w, d the d Now, s bouded beow by d sce s the u ozeo vaue of the ag weght o Thus, euaty () hods Now, we use the cocet of subcodes of geeated by as defed by V Sso ad P Soe [] The subcode of geeated by, deoted by, s the set Coyght 03 Sces OJS

5 J D PLCIO, V P SISON 3 a a geeazato of the abzzo boud was deved [] ee we ove a aae boud fo ea boc codes ove The oof eseted hee s based o the oof [] Lea 5 Let, 0 s fee f ad oy f Poof: Let be fee the the euato a 0 has oy the tva souto I atcua, ab 0 a b, that s, a b es a b Thus, Let The fo ay ozeo a ad b, a b a b That s, a b 0 a b ut geeates by defto So, s fee The et stateet s a dect coseuece of the cadaty of the deas of Cooay 55 Let The uv \ 0 f ad oy f ; f S u v \ uv o v \ uv ; uv, \ S u v v f ad oy f f ad oy 3 whee Theoe 55 (abzzo-tye oud) Let be a u ag weght codewod The Moeove, f d (5) s fee, the 3 d Poof: Let be a u ag weght codewod the cosde subcode Let deote the u ag dstace of The u ag dstace of s st d sce s a subcode of so s a subcode of wth The effectve egth of s d cog fo the d oze o ostos Dect acato of the abzzo bou d esuts to euaty (5) hods y Le a 5, euaty (6) foows 6 Eae Cosde the fee ate- sef-othogoa code ove geeated by G v uv uuv If G I the I, D, E 0, 0 ad 0 0 codewod ethe has hoogeeous weght 0, o 8 The u ag dstace of s The bay age of was obta ed wth esect to the bass u v uv, v uv, u uv,u v (6) Tabe Coaso of bouds fo Sgeto-tye 8 3 Pot-tye as-tye abzzo-tye Usg Cooay, euvaet to G s eutato The age code has a u ag dstace of 8 ad s sef-othogoa I Tabe, w e ca see that eets the ue boud of the Pot-tye ad abzzo-tye boud EEENCES [] S T Doughety, M K Guta ad K Shooto, O Geeazed Weghts fo Codes ove te gs, et, 00 [] T ood, Chaactezato of te obeus gs, chv de Matheat, Vo 0, No 6, 00, 06-5 do:0007/pl [3] S Kaadez ad Ydz, v-costacycc Codes ove u v uv, Joua of the a Isttute, Vo 37, 0, [] V Sso ad P Soè, ouds o the Mu oogeeous Dstace of the -ay Iage of Lea oc Codes ove the Gaos g G,, IEEE tasactos o Ifoato Theoy, Vo 53, No 6, 007, do:0 09/TIT [5] X Xu ad X Lu, O the Stuctue of Cycc Codes ove u v uv, Wuha Uvesty Joua of Natua Sceces, Vo 6, No 5, 0, do:0007/s [6] Ydzad S Kaadez, Lea Codes ove u v uv, Desgs Codes Cytogahy, Vo 5, No, 00, 6-8 do:0007/s [7] Ydz ad S Kaadez, Sef-dua Codes ove u v uv, Joua of the a Isttute, Vo 37, 00, No 0, do:006/fa [8] Ydz ad S Kaadez, Cycc Codes ove u v uv, Desgs Codes Cytogahy, Vo 58, No 3, 0, -3 do:0007/s Coyght 03 Sces OJS