Improved Probabilistic Bounds on Stopping Redundancy

Transcription

1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 4, APRIL The vesgao was caed ou by fomulag he poblem a dscee fom. Ths leads o he epeseao of a quadphase code by dscee Foue asfom ad vese dscee Foue asfom. The coeffces of a msmached fle ae he calculaed fom (7) by explog he dscee Foue asfom epeseao of he code. The msmached fle desged hs mae elmaes he sdelobes as we have show Fgs., 2, ad 6. A msmached fle has hee dawbacks whe compaed o he mached fle. The fs dawback s ha a msmached fle has fely may coeffces. Howeve, we have explaed ad also show Fg. 3 ha he values of hese coeffces go o zeo apdly. Hece, hs may o be poblemac pacce. The secod dawback s ha he dscee Foue asfom of he code mgh have zeos he fequecy doma ad we have show ha hs s mpobable. The hd poblem s ha s oupu SNR s less ha he oupu SNR of he especve mached fle. Howeve, oe ca choose codes wh mmum SNR losses. We have peseed he ose ehaceme facos fo he bes codes Table I. These codes have bee seleced fom umbe of vesgaed codes based o he k vs values. The quadphase codes, whch have smalle values of k vs ha he especve bay-phase codes, wee seleced. Fo code leghs of 7; 0; 9; 5; 6; 7; 8; ad 9, he opmal codes ued ou o be quadphase codes. The ohe opmal codes ae bay-phase codes. The 3- ad 5-eleme codes have 0236 ad 0302 ose ehaceme facos, especvely. Fo hese codes we have show ha he shape of he msmached fle esembles he mached fle. I mos of he ohe codes, he ose ehaceme faco s geae ha 04. Oe should evaluae he losses SNR agas he advaages of elmag sdelobes. Fally, by cayg ou a adom seach we have show ha a adomly seleced log code wll mos lkely have lage ose ehaceme faco. [] S. W. Golomb ad R. A. Scholz, Geealzed bake sequeces, IEEE Tas. If. Theoy, vol. IT-, o. 4, pp , Oc [2] R. J. Tuy, Fou-phase bake codes, IEEE Tas. If. Theoy, vol. IT-20, o. 3, pp , May 974. [3] J. W. Taylo, J. ad H. J. Blchkoff, Quadphase code-a ada pulse compesso sgal wh uque chaacescs, IEEE Tas. Aeosp. Eleco. Sys., vol. 24, o. 2, pp , Ma [4] W. H. Mow, Bes quadphase codes up o legh 24, Eleco. Le., vol. 29, pp , 993. [5] H. D. Lüke, Msmached fleg of peodc quadphase ad 8-phase sequeces, IEEE Tas. Commu., vol. 5, o. 7, pp , Jul [6] R. Coua ad D. Hlbe, Mehode de mahemasche Physk. Bel, Gemay Spge-Velag, 968. Impoved Pobablsc Bouds o Soppg Redudacy Jusheg Ha, Sude Membe, IEEE, Paul H. Segel, Fellow, IEEE, ad Alexade Vady, Fellow, IEEE Absac Fo a lea code, he soppg edudacy of s defed as he mmum umbe of check odes a Tae gaph T fo such ha he sze of he smalles soppg se T s equal o he mmum dsace of. Ha ad Segel ecely poved a uppe boud o he soppg edudacy of geeal lea codes, usg pobablsc aalyss. Fo mos code paamees, hs boud s he bes cuely kow. I hs coespodece, we pese seveal mpovemes upo hs boud. Idex Tems Bay easue chael, eave decodg, lea codes, soppg edudacy, soppg ses. REFERENCES [] R. H. Bake, Goup sychozg of bay dgal sysems, Commucaos Theoy. Lodo, U.K. Buewoh, 953, pp [2] R. J. Tuy, Sequeces wh small coelao, Eo Coecg Codes, H. B. Ma, Ed. New Yok Wley, 968, pp [3] J. Lde, Bay sequeces up o legh 40 wh possble auocoelao fuco, Eleco. Le., vol., pp , 975. [4] Coxso ad Russo, Effce exhausve seach fo opmal-peak-sdelobe bay sequeces, IEEE Tas. Aeosp. Eleco. Sys., vol. 4, o., pp , Ja [5] E. L. Key, E. N. Fowle, ad R. D. Haggay, A Mehod of Sdelobe Suppesso Phase Coded Pulse Compesso Sysems, MIT, Lcol Lab., Lexgo, MA, Tech Rep. 209, Nov [6] H. Rohlg ad W. Plagge, Msmached-fle desg fo peodcal bay phased sgals, IEEE Tas. Aeosp. Eleco. Sys., vol. 25, o. 6, pp , Nov [7] M. S. Lehe, B. Dame, ad T. Nygé, Opmal bay phase codes ad sdelobe-fee decodg fles wh applcao o cohee scae ada, A. Geophyscae, vol. 22, pp , [8] M. S. Lehe ad I. Häggsöm, A ew modulao pcple fo cohee scae measuemes, Rado Sc., vol. 22, pp , 987. [9] J. Ruppech ad M. Rupf, O he seach fo good apeodc bay veble sequeces, IEEE Tas. If. Theoy, vol. 42, o. 5, pp , Sep [0] J. Ruppech, Maxmum-Lkelhood Esmao of Mulpah Chaels, Ph.D. dsseao, Swss Fedeal Isue of Techology, Zuch, Swwzelad, 989. I. INTRODUCTION The soppg edudacy of a bay lea code chaacezes he complexy, measued ems of he mmum umbe of check odes, of a Tae gaph fo such ha eave decodg of hs gaph o he bay easue chael (BEC) acheves pefomace compaable (up o a cosa faco) o ha of maxmum-lkelhood decodg. I s wdely beleved [6], [9] ha soppg edudacy ad elaed coceps ae of elevace fo chaels ohe ha he BEC as well. Specfcally, le be a lea code of legh, dmeso k, ad mmum dsace d, ad le H [h j ] be a 2 pay-check max fo. The coespodg Tae gaph T (H) fo s a bpae gaph wh vaable odes ad check odes such ha he h check ode s adjace o he jh vaable ode ff h j 6 0.Asoppg se S s Mauscp eceved July 27, 2007; evsed Decembe 3, Ths wok was suppoed pa by he Naoal Scece Foudao. J. Ha ad P. H. Segel ae wh he Cee fo Magec Recodg Reseach, Uvesy of Calfoa Sa Dego, La Jolla, CA USA (e-mal ha@cs. ucsd.edu; psegel@ucsd.edu). A. Vady s wh he Depames of Eleccal Egeeg ad Compue Scece, Uvesy of Calfoa Sa Dego, La Jolla, CA USA (e-mal vady@klmajao.ucsd.edu). Commucaed by T. Ezo, Assocae Edo fo Codg Theoy. Dgal Objec Idefe 0.09/TIT Thoughou hs coespodece, a pay-check max fo should be udesood as ay max H whose ows spa he dual code. Thus ca be (ad ofe wll be) scly geae ha 0 k /$ IEEE

2 750 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 4, APRIL 2008 a subse of he vaable odes T (H) such ha all he check odes ha ae eghbos of S ae coeced o S a leas wce. Equvalely, a soppg se S s a se of colums of he pay-check max such ha he coespodg colum submax of H does o coa a ow of wegh oe. I s well kow [3], [9] ha eave decodg o he BEC succeeds ff he se of eased posos does o coa a soppg se. The sze of he smalles soppg se, kow as he soppg dsace s(h), s hus a lm o he umbe of easues ha eave decodg o he Tae gaph T (H) ca guaaee o coec. Noe ha he soppg dsace s o a popey of he code self, bu ahe of he specfc choce of a pay-check max H fo. Moeove, s kow [9] ha s(h) d fo all possble choces of H, ad s always possble o fd a pay-check max fo such ha hs boud holds wh equaly. Ths movaes he followg defo. Defo Le be a lea code wh mmum Hammg dsace d. The he soppg edudacy of s defed as he he smalles ege such ha hee exss a pay-check max H fo wh ows ad s(h) d. Soppg edudacy was oduced by Schwaz ad Vady [8], [9]. I was subsequely suded a umbe of ece papes [], [4], [6] [3], [7] ad [20]. Relaed coceps, such as soppg edudacy heachy, appg edudacy, ad geec easue-coecg maces wee vesgaed [8] [3], [6], ad [20]. Exsg esuls o soppg edudacy ae of wo ypes bouds o he soppg edudacy of specfc famles of codes (e.g., cyclc codes [8], MDS codes [6], [7], [9], Reed Mulle codes [4], [9], ad Hammg codes [4], [20]) as well as bouds o he soppg edudacy of geeal (bay) lea codes [6], [0] [3], [8], [9]. I s he lae ype of esuls ha we ae coceed wh hee. Thus le us descbe he kow bouds some deal. Le be a (; k; d) bay lea code, ad le 0 k. The fs uppe boud o he soppg edudacy of, esablshed by Schwaz ad Vady [8], [9, Theoem 4], s gve by d 0 2 povded d 3. Ths boud was mpoved (a leas, fo odd d) by Ha ad Segel [6, Theoem 2], yeldg whee bd2c. Hollma ad Tolhuze [0] [3] suded a moe dffcul poblem of cosucg geec easue-coecg ses. Howeve, some of he esuls ca be epeed as bouds o he soppg edudacy heachy (cf. [8], [9]) of geeal lea codes, ad he specalzed o bouds o. I pacula, Hollma ad Tolhuze pove [2, Theoem 5.2] ad [3, Theoem 4.] ha d 0 2 whch mpoves upo () ad (2). They also show [0, Theoem ] ha f all he codewods of have eve wegh, he d 0 3 () (2) (3) (4) The bouds () (4) ae all based o he same geeal se of deas, ad ae cosucve a cea well-defed sese (cf. [2], [3]). I coas, Hollma ad Tolhuze employed a ocosucve adom- TABLE I UPPER BOUNDS ON THE STOPPING REDUNDANCY OF THE (24; 2; 8) AND THE (48; 24; 2) CODES codg agume [, Theoem 4.2] ad [3, Theoem 3.2] o show ha log 2 ((2 0 )(2 0 2)( ) (2 0 2 d02 )) (5) (d 0 ) 0 log 2 (2 0 (d 0 )) As poed ou by Ludo Tolhuze, aohe pobablsc boud o soppg edudacy, amely ( 0 )(d 0 2) 0 log 2 ((d 0 2)!) + (6) (d 0 2) 0 log 2 (2 d02 0 ) follows by combg Poposo 6.2, Lemma 6.5, ad Theoem 6.4 of [2]. A eely dffee pobablsc agume was used by Ha ad Segel [6, Theoem 3] o esablsh he followg boud whee m 2 E ;d () < +( 0 d +) (7) E ;d () def 0 2 (8) Alhough he vaous bouds () (7) ae dffcul o compae aalycally, he followg clams seem o be jusfed. Fo mos code paamees, he pobablsc bouds (5) (7) ae much bee ha he cosucve bouds () (4). Amog he pobablsc bouds, he Ha Segel boud of [6, Theoem 3] ad (7) s by fa he soges. Usg he ables of [5], we have vefed ha hese clams hold fo all possble code paamees ; k, ad d wh d>4 ad 256. A epeseave pcue s show Table I, whee we compued he bouds () (7) fo he (24; 2; 8) Golay code ad fo he (48; 24; 2) quadac-esdue code. Thus he Ha Segel boud (7) appeas o be he bes cuely kow boud o he soppg edudacy of geeal lea codes. I hs coespodece, we pese seveal mpovemes upo he Ha Segel boud (7) usg a umbe of deas, mos of whch ae based upo a moe caeful pobablsc aalyss. II. PROBABILISTIC BOUNDS ON STOPPING REDUNDANCY Gve a lea code, Ha ad Segel [6, Theoem 3] cosuc a pay-check max H fo by dawg codewods depedely ad ufomly a adom fom he dual code?. Ou fs obsevao s hs such adom choce s effce a fs, bu becomes less ad less effce as successve ows of H ae daw fom?. A some po, deemsc seleco becomes supeo. Ths leads o he followg esul.

3 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 4, APRIL Theoem Le be a (; k; d) bay lea code. Le deoe he defcecy of, ha s 0 k +0 d. Le E ;d () be he expecao fuco defed (8). The TABLE II IMPROVED BOUNDS ON THE STOPPING REDUNDANCY OF THE (24; 2; 8) AND THE (48; 24; 2) CODES m 2 + be ;d ()c + (9) Poof We fs eed some oao. Le ( [] ) deoe he se of all subses of f; 2;...;g of sze, ad le def [] (0) Followg [6], [9] we wll efe o elemes of ( [] ) as -ses. Gve a m 2 bay max H ad a -se S, we say ha H coves S f he m 2 colum submax of H, cossg of hose colums ha ae dexed by he elemes of S, coas a ow of wegh oe. Clealy, he soppg ses wh espec o H ae pecsely hose ses S ha H does o cove; hece s(h) d f ad oly f H coves S fo all S2. Now le H be a 2 max whose ows ae daw fom? depedely ad ufomly a adom. If S2 s a fxed -se ad h s a fxed ow of H, he he pobably ha h coves S s 2. Ths s so because? s a ohogoal aay of segh d 0 (cf. [5, p. 39]), whch meas pecsely ha fo all S 2, a codewod daw a adom fom? s equally lkely o coa each of he 2 possble vecos o he posos dexed by S. Le X deoe he umbe of ses S2 ha ae o coveed by H. The X s a adom vaable, ad he expeced value of X s gve by [X ] S2 PfS o coveed by H g 0 2 The gh-had sde of he above expesso s E ;d (), bydefo. I follows ha hee exss a ealzao H of H ha coves all bu a mos be ;d ()c ses. Fo each ucoveed se S, hee s a codewod of? ha coves S (aga, sce? s a ohogoal aay). Thus we ca adjo be ;d ()c ows o H o ceae a max H 0 wh s(h 0 )d. I s possble ha ak(h 0 ) <0k. Howeve, ak(h 0 ) s clealy a leas d 0. Hece, by adjog a mos (0k) 0 (d 0 ) ows o H 0,wefally oba a pay-check max H 00 fo wh a mos + be ;d ()c + ows ad s(h 00 )d. Sce he mmzao (9) coas he Ha-Segel boud of (7) as a specal case, Theoem s a leas as sog as (7). I fac, s ofe subsaally soge (cf. Table II). The boud of Theoem volves solvg a mmzao poblem; a closed-fom expesso would be desable. Ths s addessed he followg coollay, whch s smla sp o [6, Coollay 4] ad mpoves upo. The coollay povdes a appoxmae closed-fom soluo fo he mmzao poblem of (9), whch becomes exac asympocally. Coollay 2 Le be a (; k; d) bay lea code. Le be he defcecy of, as befoe, ad defe C def ' 2 H D def 0 l 0 d 0 ' d whee H 2 (x) def 0x log 2 x0(0x) log 2 (0x) s he bay eopy fuco. The l C +ld + D + () Poof I ode o solve he mmzao poblem (9), albe appoxmaely, we uppe-boud E ;d () as follows E ;d () 0 d 0 2 Ce 0D whee he equaly follows fom he fac ha 2 s oceasg. Le f () +Ce 0D. The f (), egaded as a fuco fom o, s covex ad aas s global mmum a 0 (l C +ld)d. Clealy, 0 s always posve ad f ( 0 ) (l C +ld +)D. The coollay ow follows fom Theoem. Nex, we obseve ha he adom choce mehod used by Ha ad Segel [6, Theoem 3] ad Theoem s o opmal. I would be bee o selec he ows of H fom? f0g, ahe ha?, whou eplaceme, ahe ha wh eplaceme. Ths leads o he followg boud. Theoem 3 Le be a (; k; d) bay lea code, le be he defcecy of, ad le 0 k. The whee m 2 F ;d;k () def + bf ;d;k ()c + (2) j j (3) Poof We cosuc a pay-check max fo he same way as Theoem, excep ha he al max H s seleced dffeely. Specfcally, he ows h ;h 2;...;h of H ae seleced as follows. Suppose ha we have aleady chose he fs j 0 ows h ; h 2 ;...;h j0, he he jh ow h j s seleced ufomly a adom fom? f0;h ;h 2 ;...;h j0 g (4) Now le S2 be a fxed -se ha s o coveed by ay of h ; h 2 ;...;h j0. Wha s he pobably ha he jh ow coves S? The oal umbe of possble (equally lkely) choces fo h j s 2 0 j. Of hese, pecsely 2 0 cove S. Ths s so because hee ae exacly 2 0 codewods? ha cove S (aga, sce? s a ohogoal aay), ad oe of hese s amog h ;h 2;...;h j0 by assumpo.

4 752 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 4, APRIL 2008 Hece, he pobably ha h j coves S s 2 0 (2 0 j). Thus, fo ay fxed -se S2, he pobably ha H does o cove S s PfS s o coveed by H g j j Thus, F ;d;k () (3) s he expeced umbe of ses S2 o coveed by H, ad he heoem follows. ha ae Sce (3) s exemely close o (8), s somewha supsg ha Theoem 3 yelds ay mpovemes upo Theoem. Bu does, a leas fo small code paamees (cf. Table II). We ow mpove he cosuco of a pay-check max fo descbed Theoem 3 ye aohe espec. Le H 0 be a fxed 2 max whose ows h ;h 2 ;...;h ae ozeo codewods of?, ad le 0 deoe he subse of he se (0) cossg of hose ses ha ae o coveed by H 0. Le X 0 j 0j. Suppose we adjo aohe ow h + o H 0, seleced ufomly a adom fom? f0;h ;h 2;...;h g as (4), ad le H 0 deoe he esulg adom max. Le X 0 be he umbe of ses S 2 o coveed by H. 0 The, agug as he poof of Theoem 3, we fd ha [X 0 ] S2 X 0 PfS s o coveed by h +g (d 0 )20d ( +) I follows ha hee exss a ( +)2 ealzao H of H 0 ha coves all bu a mos (d 0 )20d+ X X 0 0 (5) 2 0 ( +) ses. The pocess ca be ow eaed. Tha s, gve H, we ca cosuc a ( +2)2 max H 2 ha coves all bu a mos (d 0 )20d+ X 2 X 0 (6) 2 0 ( +2) ses. Ad so o. To fomalze hs pocess, le us defe fo all j ; 2;..., he fuco P j! as follows P j (k) def (d 0 )20d+ k ( + j) ; fo all k 2 whee he paamees d;, ad ae egaded as cosa. The, afe eaos of he foegog pocedue, we wll cosuc a ( + ) 2 max H ha coves all bu a mos Q (X 0 ) def P P 0 P 2 P (X 0 ) (7) ses.ifq (X 0 )0, we ae doe. Ths esablshes he followg uppe boud o soppg edudacy. Theoem 4 Le be a (; k; d) bay lea code, wh defcecy. The he soppg edudacy s a mos m 2 + mf 2 Q (bf ;d;k ()c) 0g + (8) whee F ;d;k () s he expecao fuco defed (3) whle Q ( ) s he fuco defed (7). Alhough he defo of Q ( ) (7) appeas o be ahe volved, we obseve ha he mmzao ove (8) s, fac, vey easy o compue all akes s a sgle-le whle loop. The boud of Theoem 4 clealy cludes Theoem 3 as a specal case, ad mpoves upo (cf. Table II). Fally, we would lke o ge d of he small, bu aoyg, defcecy em Theoems 4. To hs ed, he followg smple obsevao ofe suffces. A code 2 s sad o be maxmal f s o possble o adjo ay veco 2 o whou deceasg s mmum dsace. Poposo 5 Le be a (; k; d) bay lea code. Le H be ay max wh s(h) d whose ows ae codewods of?.if s maxmal, he ak(h) 0 k. Poof Assume o he coay ha ak(h) <0k. The H s a pay-check max fo a pope supecode 0 of. Sce s(h) d, he mmum dsace of 0 s also d, whch coadcs he fac ha s maxmal. Poposo 5 mples ha f s maxmal, we ca dop he defcecy em Theoems 4. We ca also ge d of hs em usg a moe elaboae pobablsc agume. I s uvely clea ha f we daw suffcely may codewods ufomly a adom fom he dual of a (; k; d) code, he esulg max s lkely o have ak close o 0 k. Ths obsevao s made fomal he followg lemma. Lemma 6 Le be a (; k; d) bay lea code, ad le H be a 2 max whose ows ae daw ufomly a adom (ehe wh o whou eplaceme) fom he dual code?. Le 0 k, ad defe he adom vaable Y 0 ak(h ). The fo all, we have [Y ] (9) Poof Skech Fs assume ha he ows of H ae daw fom? ufomly a adom wh eplaceme (as Theoem ). I s kow (see [4], fo example) ha Pfak(H )jg j0 2 (0j)(0j) 0 ( )( ) 0 2 0j fo all j ; 2;...;. Takg he expecao wh espec o hs dsbuo (alog wh some echcal deals ha we om) poduces he uppe boud (9). Fally, ca be show ha f he ows of H ae daw ufomly a adom fom? f0g whou eplaceme (as Theoem 3), hs ca oly educe he expeced value of Y. I ode o combe Lemma 6 wh ou eale esuls, le us defe he fuco G ;d;k () def F ;d;k () (20) whee 0 k ad F ;d;k () s as defed (3). We ca ow modfy he poofs of Theoem 3 ad Theoem 4 accodgly, heeby esablshg he followg esul. Theoem 7 Le be a (; k; d) bay lea code, ad le 0 k. The he soppg edudacy of s bouded by m Moeove, f ( 0 )(d 0 ) m + bg ;d;k ()c (2) 2 he + mf 2 Q (bg ;d;k ()c) 0g (22) whee he fucos G ;d;k () ad Q ( ) ae as defed (20) ad (7), especvely.

5 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 4, APRIL Poof Use he same agume as he poof of Theoem 3, bu wh espec o he adom vaable Z X + Y, whch s he sum of he umbe of ses S2 o coveed by H ad s ak defcecy 0 ak(h ). Fuhe, oe ha he codo ( 0 )(d 0 ) 2 s suffce fo he agume (5), (6), ad (7) o be applcable hs case as well. I s o mmedaely appae ha (2) poduces a bee boud o soppg edudacy ha Theoem 3. Howeve, we ca show ha hs s always so, excep fo a few val cases. I fac, compag (3) ad (9), we see ha he secod em (20) deceases wh expoeally fase ha he fs em. Thus, uless he mmum (2) ad/o (22) s acheved fo vey close o ( whch case mus be close o as well), he secod em (20) has esseally o effec o he mmzao hs em s a y faco, whch dsappeas whe akg he floo of G ;d;k (). I follows ha fo vually all code paamees, he e effec of (2) ad (22) cosss of elmag he defcecy em Theoems 3 ad 4. III. DISCUSSION AND CONCLUDING REMARKS The Ha-Segel pobablsc boud [6, Theoem 3] s he bes cuely kow boud o he soppg edudacy of geeal lea codes. We peseed seveal mpovemes upo hs boud based upo a moe elaboae pobablsc aalyss. Alhough all ou esuls wee saed ad poved hee fo bay lea codes, hey exed saghfowadly o lea codes ove a abay fe feld. Exeso o bouds o he soppg edudacy heachy [8], [9] s also saghfowad smply eplace all occueces of d Theoems 4 ad 7 by he coespodg dex of he heachy. Geealzao o bouds o he appg edudacy [6] should be elavely easy as well, alhough, pehaps, less saghfowad. Whle he mpovemes o he Ha Segel boud esablshed hee ae o damac, we beleve s useful o have he soges possble fom of hs boud avalable he leaue. We po ou ha, addo o he esuls peseed Seco II, we have vesgaed seveal ohe deas. Fo example, oe could cosuc a pay-check max fo a lea code by selecg each codewod of he dual code? depedely wh some pobably p, ad he opmze he value of p. Howeve, we have foud ha whle hs mehod, as well as ohe pobablsc-choce vaas, mpove upo he ogal Ha Segel boud of [6, Theoem 3], hey ae geeally less effce ha he bouds peseed Seco II. We oe ha well-kow echques pobablsc mehod, such as Lovász local lemma [2, p. 64] ad Rödl bble [2, p. 53], do o seem o be applcable ou coex. Thus we beleve ha fuhe mpovemes, f ay, would eque dascally ew deas. [3] C. D, D. Poe, I. Telaa, T. J. Rchadso, ad R. L. Ubake, Fe-legh aalyss of low-desy pay-check codes o he bay easue chael, IEEE Tas. If. Theoy, vol. 48, o. 6, pp , Ju [4] T. Ezo, O he soppg edudacy of Reed-Mulle codes, IEEE Tas. If. Theoy, vol. 52, o., pp , Nov [5] M. Gassl, Bouds o he Mmum Dsace of Lea Codes, Jul. 2, 2007 [Ole]. Avalable a [6] J. Ha ad P. H. Segel, Impoved uppe bouds o soppg edudacy, IEEE Tas. If. Theoy, vol. 53, o., pp , Ja [7] J. Ha, P. H. Segel, ad R. M. Roh, Bouds o sgle-excluso umbes ad soppg edudacy of MDS codes, Poc. IEEE I. Symp. Ifomao Theoy, Nce, Face, Ju. 2007, pp [8] T. Heh, S. Laede, O. Mlekovc, ad J. B. Hube, The soppg edudacy heachy of cyclc codes, Poc. 44h Au. Alleo Cof. Commucaos, Cool ad Compug, Mocello, IL, Sep. 2006, pp [9] T. Heh, O. Mlekovc, S. Laede, ad J. B. Hube, Pemuao decodg ad he soppg edudacy heachy of lea block codes, Poc. IEEE I. Symp. Ifomao Theoy, Nce, Face, Ju. 2007, pp [0] H. D. L. Hollma ad L. M. G. M. Tolhuze, Geeag pay check equaos fo bouded-dsace eave easue decodg of eve wegh codes, Poc. 27h Symp. Ifomao Theoy he Beelux, Noodwjk, The Nehelads, Ju. 2006, pp [] H. D. L. Hollma ad L. M. G. M. Tolhuze, Geeag pay check equaos fo bouded-dsace eave easue decodg, Poc. IEEE I. Symp. Ifomao Theoy, Seale, WA, Jul. 2006, pp [2] H. D. L. Hollma ad L. M. G. M. Tolhuze, Geec easue-coecg ses bouds ad cosucos, J. Comb. Theoy, Se. A, vol. 3, pp , Nov [3] H. D. L. Hollma ad L. M. G. M. Tolhuze, O pay check collecos fo eave easue decodg ha coec all coecable easue paes of a gve sze, IEEE Tas. If. Theoy, vol. 53, o. 2, pp , Feb [4] I. N. Kovaleko, O he lm dsbuo of he umbe of soluos of a adom sysem of lea equaos he class of boolea fucos, Theoy Pobab. App., vol. 2, pp , 967. [5] F. J. MacWllams ad N. J. A. Sloae, The Theoy of Eo-Coecg Codes. Amsedam, The Nehelads Noh-Hollad, 978. [6] O. Mlekovc, E. Solja, ad P. Whg, Soppg ad appg ses geealzed coveg aays, Poc. 40h Cof. Ifomao Scece ad Sysems, Pceo, NJ, Ma. 2006, pp [7] E. Roses ad Ø. Yehus, A algohm o fd all small-sze soppg ses of low-desy pay-check maces, Poc. IEEE I. Symp. Ifomao Theoy, Nce, Face, Ju. 2007, pp [8] M. Schwaz ad A. Vady, O he soppg dsace ad he soppg edudacy of codes, Poc. IEEE I. Symp. Ifomao Theoy, Adelade, Ausala, Sep. 2005, pp [9] M. Schwaz ad A. Vady, O he soppg dsace ad he soppg edudacy of codes, IEEE Tas. If. Theoy, vol. 52, o. 3, pp , Ma [20] J. H. Webe ad K. A. S. Abdel-Ghaffa, Soppg se aalyss fo Hammg codes, Poc. IEEE If. Theoy Wokshop, Rooua, New Zealad, Aug. 2005, pp ACKNOWLEDGMENT The auhos ae gaeful o Makus Gassl ad Ludo Tolhuze fo he help wh efeeces [5] ad [0] [3], especvely. REFERENCES [] K. A. S. Abdel-Ghaffa ad J. H. Webe, Complee eumeao of soppg ses of full-ak pay-check maces of Hammg codes, IEEE Tas. If. Theoy, vol. 53, o. 9, pp , Sep [2] N. Alo ad J. H. Spece, The Pobablsc Mehod. New Yok Wley, 99.