QPSK Block-Modulation Codes for Unequal Error Protection

Transcription

1 San Jse State University SJSU SchlarWrks Faculty Publicatins Electrical Engineering 199 QPSK Blck-Mdulatin Cdes fr Unequal Errr Prtectin Rbert H. Mrels-Zaragza Osaka University, Shu Lin University f Hawaii at Mana Fllw this and additinal wrks at: Part f the Electrical and Cmputer Engineering Cmmns Recmmended Citatin Rbert H. Mrels-Zaragza and Shu Lin. "QPSK Blck-Mdulatin Cdes fr Unequal Errr Prtectin" Faculty Publicatins (199): -1. di:.19/1.01 This Article is brught t yu fr free and pen access by the Electrical Engineering at SJSU SchlarWrks. t has been accepted fr inclusin in Faculty Publicatins by an authrized administratr f SJSU SchlarWrks. Fr mre infrmatin, please cntact schlarwrks@sjsu.edu.

2 EEE TRANSACTONS ON NFORMATON THEORY. VOL. 1, NO.2. MARCH 199 2) in the case where i '" j when T = (pm + 1)/. fr () SS pm - 2 ccurs (p,,- l + pm _ 2p"'-') times ccurs (pn-l _ 2pm-l) times. fr l S ' ::;p - l. V. CONCLUSON t was shwn that the new family cnsisting f pn/ 2 (where n is even) balanced nnbinary sequences with perid pn - 1 can be btained frm the mdified Kumar-Mren sequences f the same perid, and the distributin f crrelatin values fr the family was shwn t have 1'+ 2 distinct crrelatin values and the same maximum nntrivial crrelatin value f 1',,/ as that f Kumar-Mren sequences. On the ther hand, it was shwn that the cst f making sequences balanced is a decrease f family size in additin t the cnditin that n is an even number. The family size f the new sequences is pn/2 which is much smaller than p", that f Kumar-Mren sequences. V. ACKNOWLEDGMENT The authrs wish t thank Prf. P. V. Kumar fr sending the draft f their paper [] t ne f the authrs (K..), and wuld like t thank the annymus referees fr helpful cmments useful fr imprving the readahility f the paper. REFERENCES [] D. V. Sarwate and M. B. Pursley, "Crss-crrelatin prperties fpseudrandm and related sequences." Prc. EEE, vl., pp. 9-1, May 190. [2] M. K. Simn, 1. K. Omura, R. A. Schltz, and B. K. Levitt, Spread Spectrum Cmmunicatins, vl. 1. Cmputer Science Press, 19, ch.. [] P. V. Kumar and O. Mren, "Prime-phase sequences with peridic crrelatin prperties better than binary sequences," EEE Trans. nfrm. Thery, vl., n., pp. 0-1, May [] L. R. Welch. "Lwer bunds n the maximum crrelatin f signals," EEE Trans. nfrm. Thery, vl. 1T-20, pp. 9-99, May 19. S R. Lidl and H. Niederreiter, "Finite fields," in Vl. 20 f Encyclpedia f Mathematics and ts Applicatins. Amsterdam, The Netherlands: Addisn-Wesley, 19. [] S. Matsufuji, K. mamura, and S. Sejima, "Balanced binary pseudrandm sequences with lw p<:rimlic crrelatiun," presented at the 1990 EEE nt. Symp. n nfrmatin Thery, Jan QPSK Blck-Mdulatin Cdes fr Unequal Errr Prtectin Rbert H. Mrels-Zaragza, Member, EEE, and Shu Lin, Fellw, ieee Abstract- Unequal errr prtectin (UEP) cdes find applicatins in bradcast cbannels, as well as in tber digital cmmunicatin systems, where messages bave ditterent degrees f imprtance, n this crrespndence, binary linear UEP (LUEP) cdes cmbined with a Gray mapped QPSK signal set are used t btain new efficient QPSK blck-mdulatin cdes fr unequal errr prtectin. Several examples f QPSK mdulatin cdes that have tbe same minimum squared Euclidean distance as the best QPSK mdulatin cdes, f the same rate and length, are given. n the new cnstructins f QPSK blck-mdulatin cdes, even-length binary LUEP cdes are used. Gd even-length binary LUEP cdes are btained wben shrter binary linear cdes are cmbined using either the well-knwn lulu + vi-cnstructin r the s-called cnstructin X, Bth cnstructins have the advantage f resulting in ptimal r near-ptimal binary LUEP cdes f shrt t mderate lengths, using very simple linear cdes, and may be used as cnstituent cdes in the new cnstructins. LUEP cdes lend themselves quite naturally t multistage decdings up t tbeir minimum distance, using the decdings f cmpnent subcdes. A new subptimal tw-stage sft-decisin decding f LUEP cdes is presented and its applicatin t QPSK blck-mdulatin cdes fr UEP illustrated. ndex Terms-Unequal errr prtectin, cded mdulatin, multistage decding.. :'ltroducto:'l There are many practical applicatins in which it is required t design a cde that prtects messages against different levels f nise, r messages with different levels f imprtance ver a nisy channel f the same nise pwer level. Examples f such situatins are: bradcast channels, multiuser channels, cmputer netwrks, pulsecded mdulatin (PCM) systems and surce-cding systems, amng thers. Such a cde is usually said t be an unequal errr prtectin (VEP) cde. n this crrespndence, we prpse t use binary linear VEP (LVEP) cdes [1], cmbined with Gray mapped QPSK signal cnstellatins, t btain new efficient QPSK blck-mdulatin cdes with unequal squared Euclidean distances. That is, cde sequences assciated with the mst imprtanl message bits are separated hy a squared Euclidean distance (SED) larger than the SED between cde sequences assciated with less imprtant message bits. Several examples f LVEP QPSK blck-mdulatin cdes, having the same minimum squared Euclidean distance (MSED) as that f ptimal QPSK mdulatin cdes f the same rate and length [21. [1. are given. The crrespndence is rgauized as fllws. u Sectin, basic cncepts and tw cnslructins f LVEP cdes based n specifying the generatr matrix are presented. Sectin deals with new cnstructins f QPSK blck-mdulatin cdes and intrduces a new subptimal tw-stage sft-decisin (TSD) decding f LVEP Manuscript received February 1, 199. This research was supprted by the NSF under Grants NCR-10, NCR-910, by NASA under Grant NAG -91, and by the Japanese Sciety fr the Prmtin uf Science. D n. 91. Part f this wrk was presented at the 199 EEE nternatinal Sympsium n nfrmatin Thery, San Antni, TX, January 19, 199. R. M. Mrels-Zaragza is with the Department uf nfurmatin and Cmpuler Sciences, Faculty f Engineering Science. Osaka University, Tynaka, Osaka 00, Japan. S. Lin is with the Department f Electrical Engineering, University uf Hawaii, Hlmes Hnlulu, H 922 USA. EEE Lg Number $ EEE

3 EEE TRANSACTONS ON NFORMATON THEORY. VOL. 1. NO.2. MARCH 199 cdes. An example is given which illustrates TSD decding f QPSK blck-mdulatin cdes fr UEP. Finally, in Sectin V, cnclusins n the results are presented.. BASC CONCEPTS OF LUEP CODES When a cde is used t prvide multiple levels f errr prtectin, the cnventinal definitin f minimum distance must be generalized. Since different levels f errr prtectin are pssible with a UEP cde, a vectr f minimum distances, ne fr each level f errr prtectin, needs t be defined. Let C be an (n. h') blck cde (nt necessarily linear) ver a finite alphabet..t, n 2: '. That is, C is a ne-t-ne mapping frm 11' t.-1", i.e. where,-' >-; c(m) EA." A = A x A x... x.-1, -- k times As usual, an element m frm..t k is called a message, and an element c( fl.) frm C is called a cdewrd,.-1' is knwn as the message set. Let A k be decmpsed int the direct prduct f tw disjint message subsets,.-1"', i = 1,2, such that..t k = A" x.-1 "2, A message m E.1 k can then be expressed as m = (m,.m2). mi E _-1Ai, i = 1,2 where each m, is called the ith message part, i = 1, 2, The separatin vectr f C is defined as the tw-tuple ii = (,,2), where i = 1. 2 and r(x, x ' ) dentes the Hamming distance between x and x ' in A", Nte that in the definitin f Si abve, there is n restrictin n mj, m, fr j # i. Assume that C has bth cmpnents f its separatin vectr distinct and arranged in decreasing rder, i.e., s, > S2. such that C is an (n. ) blck cde f minimum distance S2. We call m the mst imprtant message part and m 2 the least imprtant message part. Cde C is said t be an (, 1;) tw-level UEP cde f separatin vectr ii = (1,"), fr the message set A k] X A k,. This crrespndence cncentrates n hinary linear tw-level errr crrecting cdes. That is,.-1. = {O, }. Fr a binary linear tw-level errr crrecting cde, r binary LUEP cde C, each element f the separatin vectr is given by f, min{wt(c(m)): m, # O. m, E {O. 1 }"}. i = L 2 () where wt (x) dentes the Hamming weight f vectr x. C is called an (,k) tw-level LCEP cde, f separatiun vectr s = (Sl,2), fr the message space {D. }" x {O, 1J'2, A, LUcP Cdes Specified by their Generatr Matrix n this subsectin, cnstructins f LUEP cdes by appending esets f subcdes in binary linear cdes are presented. These cnstructins may be used t btain cnstituent binary LUEP cdes which, in cnjunctin with Gray mapped QPSK signals, yield efficient QPSK blck-mdulatin cdes fr unequal errr prtectin (see Sectin ). 1) The lulu + vi Cnstructin: Fr i 1,2, let C, be an (l.k"d,) binary linear cde with generatr matrix Gi. Define the append peratin between tw vectrs, as U=(10.1],',ln_l) and V=(l',"" ",l'n-,) ilv (llo,'l1 Un-,''O,1'1,.Vn Based n C1 and C2 the fllwing cde: fl(c,.c2) = {wi w = u 0 (u + v),u E C.v E Cd is a (211. " + 1;2) binary linear cde with generatr matrix and minimum distance d = mill{2dl, max{dl.d2}} [9]. Therem.' dc" C2) is a tw-level binary LUEP cde f separatin vectr S = (Sl,S2), fr the message space {0.1 }k x {a. 1 }k2, where and 1 = min{2dl, max{ d" d2}} S2 = min{max{rll.rl2},d2} d2 = Prf' Sec []. 2) Cnstructin X: Fr i 1. 2,, let C; dente a linear (l,.k"d,) binary cde, Assume C C2, s that k. '2 and d 2: d2. Let Cx be the linear cde whse generatr matrix is Gx (G\ G2) = G" where G, [G! G!lT and G arc the generatr matrices f C, C2, and Cl, respectively. (Nte that it is required that ' = k2 - k.) Then Cx is an (ll + fl, "1 + k) linear cde f minimum distance dx = min {d. d + dl [9]. This methd f cmbining shrter linear cdes t btain a linear cde f increased length and minimum distance is knwn as Cnstructin X [], and can be viewed as a generali atin f the lulu + vi cnstructin. By an argument similar t that used t prve Therem, we can prve the fllwing Therem 2. Therem 2: C},- is a w-level binary LUEP cde f separatin vectr s = ('.. 2), fr the message space Al = {O. } k, X {a, 1 }k, whcre and. LUEP QPSK MODULATON CODES n this sectin, a methd is presented fr cmbining binary tw-level LUEP cdes with a QPSK signal set t achieve cded mdulatin schemes that ffer tw values f minirrzum squared Euclidean distance, ne fr each message part t be prtected. n ther wrds, symbls f the mst imprtant message part are mapped nt cde sequences with a larger squared Euclidean distance (SED) between them than the SED between cde sequences crrespnding t the less imprtant message part. With data transmissin ver an additive white Gaussian nise (AWGN) channel, and a gd mdulatin cde (i.e., efficient sft-decisin decding and small number f nearest neighbrs), a smaller prbability f bit errr is achieved fr the mst imprtant message part than fr the rest f the message. T apprach the errr perfrmance given by a minimum 1)'

4 EEE TRANSACTONS ON NFORMATON THEORY. VOL. 1, NO.2, MARCH 199 Fig.. 2 () 0 (00) () A QPSK signal cnstellatin with Gray mapping. 2n 1 1 k TABLE SOME LUEP QPSK BLOCK-MODULATON CODES k, k, R(bits/dim) G,(dS) G,(dB) 1/ / / / / * /. 0. / / /2.01 "* 1/ /.9 / / /2..01 * 1/ /. 11/ / /.1 1. squared Euclidean distance (MSED), a new subptimal tw-stage sft-decisin decding f tw-level LUEP cdes, that emplys their trellis structure, is intrduced. * = LUEP QPSK cde based n the lulu + vi cnslructin. A. Cnstructins via Gray Mapping n a QPSK signal c'!stellatin with Gray mapping between labels and signal pints, the squared Euclidean distance between signal pints is prprtinal t the Hamming distance between their lahels. This QPSK signal cnstellatin is said t frm a secnd-rder Hamming space []. By mapping 2-bit symbls nt signal pints in a QPSK signal set, via Gray mapping, (2n, kl + /;2) tw-level LUEP cdes and QPSK signal sets are cmbined t achieve a blckcded mdulatin system that ffers tw values f minimum squared Euclidean distances, ne fr each message part. Sme f the resulting QPSK blck-mdulatin cdes will he shwn t have the same minimum squared Euclidean distance as that f ptimal QPSK blckmdulatin cdes f the same rate and length [2], [], while ffering in additin a larger minimum squared Euclidean distance between cde sequences assciated with the mst imprtant message symbls. The prpsed cunstrudiun is as fllws: Let C he a (2n. k, + k 2 ) hinary LUEP cde f separatin vectr S = (1,2) fr the message space {O, l}kj X {O, 1}k2. Let dente the label set f the unit-energy QPSK signal cnstellatin depicted in Fig. and define the fllwing Gray mapping M between tw-bit symbls and = {0.1.2.}: The set , C = M(Cb) = {(ip, <P, <Pn d : til, = M(c2,C2,+,j E S. (co,( )....C2n tl E Cl is said t be a tw-level LUEP QPSK blck-mdulatin cde f length f, dimensin k, rate n = /;/211 (bits per dimensin), and squared Euclidean separatin vectr l] SSE!) = (21.22) where, fr i = 1. 2, the ith cmpnent f SSED is defined as the minimum squared Euclidean distance (MSED) between any tw signal sequences in C whse crrespnding ith message bits differ. (n [], SSED is defined as the MSED between signal sequences whse crrespnding ith cde psitins differ.) Fig. 2. Trellis diagram fr an LUEP QPSK cde f length. Fr AWGN channels at very high signal-t-nise ratis, and given the MSED and rate f a mdulatin cde, the asympttic cding gain G is defined as the rali f the MSED f the cded system t the MSED f an uncded system transmitting at the same rate (r number f hits per signal) []. Althugh this cding gain is never realized in practical systems, it is used t prvide a measure n the imprvement in errr perfrmance f a cded system with respect t a cmparable uncded system. Accrdingly, fur each cmpnent f SSED an asympttic cding gain is assciated. n this crrespndence, the asympttic cding gain vectr is defined as where, fr i = 1, 2 2 G, [, ] = g) O sin 2 (11"/2fl) (db). T illustrate this cnstructin methd, in Table sme QPSK blckmdulatin cdes with tw levels f errr prtectin are listed. Cdes labeled with * in the rightmst clumn f Table are LUEP QPSK mdulatin cdes btained frm the lulu + vi cnstructin, have the same minimum squared Euclidean distance as that f ptimal QPSK blck-mdulatin cdes f the same rate and length [2]. [], and prvide additinal cding gain (r, equivalently, smaller prbability

5 EEE TRANSACTONS ON NFORMATON THEORY. VOL 1. NO.2, MARCH '----' ".!! CO a:. " e u:; iii Fig.. Errr perfrmance f an LUEP QPSK mdulatin cde f length i. Eb/N Fig.. (a) (h) Trellis diagrams used in tw-stage sft-decisin decding. f bit errr) fr the '" mst imprtant message bits. Other cdes are taken frm [t]. B. Tw-Stage Sft-Decisin Decding Let C be an (n. ) tw-level LUEP cde f separatin vectr S = (S" "2) fr the message space {O.l}"] x {O.l}". Then C can be represented as the direct sum f subcdes C, and l'2. l' = l', l'2. i.e C = {c = c, + C2 : (;, E (', and 2 E C'2} where C2 is an (n. 2. S2) subcde which cntains all cdewrds f minimum weight f C, and C, is an (n. k" d"""?: 1) subcde spanned by a system f cset representatives f C2 in C. Let T, be a trellis diagram fur subcde C, f l', i = 1.2. Then a trellis diagram f C' can be expressed as the direct prduct f T and T2, T = T, - n. That is, states in T are pairs (1.2). where, is a state in T" fr i = The pair (., 2) is jined t all pairs (.,;.,i). in such a way that, fr i = s, is jined t s: in T, []. The Viterbi maximum-likelihd decding algrithm can then be applied t T t estimate the mst likely cdewrd f C using sft decisins. T reduce the number f cmputatins in sft-decisin decding f a mdulatin cde. a technique called multistage decding is usually empluyed. The prupsed subptimal tw-stage sft-decisin decding fr tw-level LUEP cdes is as fllws: ) Using sft decisins (squared Euclidean distance) and the Viterbi algrithm, determine the clsest path 1'1 in T[ t the received sequence, where T{ is a trellis crrespnding t C ", C, C a supercde f C2 At this decding stage, the mst imprtant message pat1 is decded. 2) Using sft decisins and the Viterbi algrithm, determine the clsest path ( in 1'1 + T t the received sequence, t estimate the least imprtant message part. Here (' + T2 indicates that the value f ('" btained in the first decding stage, is used a each decding step f the Viterbi algrithm perating n trellis T2 This tw-stage sft-decisin decding is well knwn, see [2], [], [11]-[ 1]. Hwever, this appears t be the first time, t the best f ur knwledge. that multistage sft-decisin decding has been explicitly used fr unequal errr prtectin cdes. Althugh at each stage the decding is maximum-likelihd, the multistage sft-decisin decding methd described abve is subptimal. At each decding stage, the mst likely path is estimated using nly part (T;) f the trellis T f C. This subptimal multistage sli-decisin decding is knwn t increase the effective number f nearest neighbrs, but this results in nly a fractin f a decibel in verall cding gain reductin (see r21. rl l], []). C. An llustrative Example n this example we cnstruct an LUEP QPSK blck mdulatin cde f length, and decde it using the subptimal tw-stage sft-decisin decding described abve. Let C be a (i.. 2) paritycheck cde and C2 be a (. L ) repetitin cde. Then applying the lulu + vi cnstructin, we btain a (1.) binary LUEP

6 0 EEE TRANSACTONS ON NFORMATON TlEORY, VOL 1, NO.2, MARCH Y ,, _, !.. X:. " e w - iii. 1O.i t Fig.. EblN Errr perfrmance f tw-stage versus maximum-likelihd sft-decisin decding. cde Cb f separatin vectr s = (,), fr the message space {O, } 1 X {O. } G. With Gray mapping between 2-bit symbls and QPSK signals, we btain an LUEP QPSK cde C f length, rate R = 1/2 (bits per dimensin), and squared Euclidean separatin vectr SSED = (1,). The reference uncded system is BPSK, which has an MSED f. t fllws that the asympttic cding gain vectr fr this LUEP QPSK blck mdulatin cde is G = (...01). T btain the trellis fr cde C', the fllwing permuted versin f Cb is cnsidered: Repeat each branch f T" the trellis f cde C" twice. This is the lulul part f the cnstructin, where u E Cl. This is equivalent t substituting in T, each branch label by 00 and each branch label 1 by 11. Then mdify trellis T2 f cde C2 by appending a t each branch label, thus cnstructing the lii part f the cde, where ii E C2 n this case, this is equivalent t replacing in T2 each branch label by n and each branch label 1 by 01. The trellis f the binary LUEP cde Cb is then the direct prduct f Tl and T 2, Tl, T2, crrespnding t liiliil + lvi. with ii E Cl and v E C2. Replacing each 2-bit branch label by an element in S = {O,, 2, }, the labels fr the QPSK signal set in Fig., accrding t the Gray mapping M f Sectin -A, results in the trellis T shwn in Fig. 2. Nte that the minimum squared Euclidean distance between any path in the upper subtrellis and any path in the lwer subtrellis f Fig. 2 is 2 x = 1, while the minimum squared Euclidean distance between paths within a subtrellis is 2 x,1 =. (By a path we mean a sequence f QPSK signals whse labels are a path in the trellis). n additin, the signal labels used in a subtrellis are frm the same BPSK signal subcnstellatin, i.e., {O, 2} fr the upper subtrellis and {L } fr the lwer subtrellis. Sft-decisin decding can nw be perfrmed using the Viterbi algrithm with squared Euclidean distances as branch metrics. Cnsider maximum-likelihd sft-decisin decding. At high signal-ta-nise ratis n an AWGN channel, the prbability f a blck errr p, is dminated by the prbability f taking a path in the trellis at minimum squared Euclidean distance, and can be apprximated by p(bl 'V(d. )Q (a"fdmin) E """'.1 'Jln where N( dmin) is the number f paths in the trellis at MSED and a 2 is the average signal pwer. Fr this LUEP QPSK blck-mdulatin cde, the prbability f a blck errr depends n what message part is being cnsidered. Fr the least imprtant message part ( bits). we have 20- p ;) = 21Q (0,;2) + Q(2a) + Q (avis) while fr the mst imprtant message part ( bit), p ) = Q(0vf:.). n bth f the abve expressins, zer-mean unit-variance additive white Gaussian nise is assumed. Nte that the abve expressins are upper bunds n the prbabilities f a bit errr, P'l and P'2' in the mst and least imprtant message parts, respectively. n Fig., we plt the prbability f a bit errr fr uncded BPSK and cmpare it with cmputer results n the bit errr rate f the least imprtant message part P" and f the mst imprtant message part, P'l' The results f Fig. were btained using a ne-step maximum-likelihd sft-decisin decding the Viterbi algrithm and the trellis diagram f Fig. 2. Frm Fig., the simulated cding gains at prbability f a bit errr f - are apprximately Gj =. (db) and G = 2.2 (db). fr the mst and least imprtant bits, respectively. These numbers agree well with the expected cding gains G =.2 (db) and G; = 2.1 (db), which are btained frm the asympttic cding gain vectr and taking int accunt the effects f the number f learest neighbrs ( and 21, respectively), using the well-knwn rule f thumb [1] which states that, at prbability f a bit errr f -, dubling the number f nearest neighbrs results in abut 0.2-dB cding lss. Attentin is nw turned t tw-stage sft-decisin decding. n the first stage f decding, trellis T;, with branch labels as shwn in Fig. (a), is used t decde the mst imprtant message bit. Nte that, frm the pint f view f decding the mst imprtant bit,

7 EEE TRANSACTONS ON NFORMATON THEORY, VOL. 1. NO.2, MARCH the number f nearest neighbrs has dubled, frm fr ne-step maximum-likelihd sft-decisin decding t fr tw-stage sftdecisin decding. n the secnd decding stage we use T, mdified accrding t the decisin in the previus step. f the decded mst imprtant bit in the lirst decding step is a 0, then we use the trellis T2 shwn in Fig. (b). f the decded message bit in tbe first stage is a, then we mdify T2 replacing each branch label 0 by 1 and each branch label 2 by.. The cmputer-simulated errr perfrmance f this tw-stage sft-decisin decding (TSD) is presented in Fig,, and cmpared t ne-stage maximum-likelihd sft-decisin decding (MLD). At a bit errr rate f -' fr the mst imprtant message bit. TSD requires abut 0.1 db mre Ej N than with single-stage MLD. This is caused hy the twfld increase in the number f nearest neighbrs in the first decding stage, as mentined befre. t can be seen frm Fig. that the errr perfrmance f the secnd decding stage, p" (TSD), is very clse t that f MLD. Once a crrect decisin n the mst imprtant message bit is made, the subtrellis used in TSD t decde the least imprtant message bits is the same as in MLD. Therefre, at high E,jS, abut the same errr perfrmance is btained. These results agree with the bservatin made in [] that degradatin f verall cding gain, with tw-stage sft-decisin decding, is negligible if the MSED f trellis diagram Tl f subcde C[ is larger than the MSED f the trellis diagram T f the supercde. [] H. 1mai and S. Hirakawa, "A new multilevel cding methd using errrcrrecting cdes," EEE Trans. nfrm. Thery, vl. T-2, n., pp. 1-, May 19. [] K. Yamaguchi and H. 1m ai, "A new blck cded mdulatin scheme and its sft decisin decding," in Prc. 199 EEE nt, Symp. n nfrmatin Thery (San Antni, TX, Jan. 1-22, 199), p.. 11 F. R. Kschischang, P. G. de Buda, and S. Pasupathy. "Blck cs et cdes fr.\f-ary phase shift keying," EEE J. Selected Areas Cmmwl.. vl., n., pp. 90()"'9, Aug. 19R9. [] W. 1. Van Gils, "Linear unequal errr prtectin cdes frm shrter cudes" EEE Tram. nfrm. Thery, vul. T -0, n., pp. -, May 19. [9] 1. F. MacWilliams and N. 1. A. Slane, The Thery f Errr-Crrecting Cdes. Amslerdam, The Netherlands: Nrth-Hlland, 19. [] W. J. Van Gils, "Tw tpics n linear unequal errr prtectin cdes: Bunds n their length and cyclic cde class es," EEE Trans. nfrm. Thery, vl. T-29, nu., pp. -, Nv. 19. [11] A. R. Calderhank, "Multilevel cdes and multistage decding," EEE Trans. Cmmun., vl., n., pp , Mar [] F. Hemmati, "Clsest easel decding f lulu + vi COlleS," EEE J. Selected Areas Cmmun., vl., n., pp. 92-9, Aug [1] T. Takata, Y. Yamashita, T. Fujiwara, T. Kasami, and S. Lin, "On a subptimum decding f decmpusable bluck cdes," in Cded Mdulatin and Bandwidth-Efficient Transmissin, E. Biglieri and M. Luise, Eds. Amsterdam, The Netherlands: Elsevier, pp [1] G. Ungerbek, "Trellis-cded mdulatiun with redundanl signal sets, Part : State f the art," EEE Cmmun. Mag., vl. 2, n. 2, pp. -21, Feb. 19. V. CONCLUSONS A new cnstructin f QPSK blck-mdulatin cdes fr unequal errr prtectin f tw types f messages was intrduced. These cdes ffer tw values f minimum squared Euclidean distance (MSED) between cded signal sequences assciated with each message patt. That is, cded signal sequences assciated with the mst imprtant message part are separated by a squared Euclidean distance (SED) larger than the MSED fr the cde. Whcn these signal sequences are transmitted ver an AWGN channel, a larger SED results in a :;rrwller prbability 0/ errr fr the mst imprtant message symbls. A Gray mapped QPSK signal set was used t btain a secnd-rder Hamming space in which (2n. k) LUEP cdes f separatin vectr s = (1.2) are mapped nt (11, k) LUEP QPSK mdulatin cdes f squared Euclidean separatin SSED = (21, 2s2). Fr shrt lengths, sme f the new QPSK blckmdulatin cdes have the same cding gain as that f ptimal QPSK mdulatin cdes f the same rate and length [J. A new subptimal tw-stare sft-decisin decdinr fr LUEP cdes was presented and an illustrative example shwed its applicatin in decding QPSK blck-mdulatin cdes fr unequal errr prtectin. The results suggest that, with tw-stage sft-decisin decding f QPSK blck-mdulatin cdes fr UEP, bth cdinr Rains are reduced by nly a fractin f a decibel, in the same way that verall cding gain degrades fr cnventinal (equal errr prtectin) mdulatin cdes. REFERENCES [] B. Masnick and J. Wlf, "On linear unequal errr prtectin cdes," EEE TraM.'. nfrm. Thery, vl. T-, n., pp , July S. L. Sayegh, "A class uf ptimum blck cdes in signal space," EEE Trans. Cmmun.. vl. COM-, n., pp. -. Oct. 19R. lj], private cmmunicatin (tables f cdes frm reference abve), [] N.J. A. Slane, S. M. Reddy, and C. L. Chen, "New binary cdes." EEE Trans. nfrm, Thery, vl. T-S, n., pp. 0-, July 192. The Nnexistence f Sme Five-Dimensinal Quaternary Linear Cdes R. Daskalv and E. Metdieva Abstract---Let 11 (.'. d) be the smallest integer n, such that a quaternary linear [11. k, d; ]-cde exists. t is prved that n,(, 20) = 0, 11(, 2) :>: 9,11(,) :>:, "(,) :>:, "(, 0) = 9,11(,) :: 19, 11.,(,1) :>: 19,1(, Hi) :>: 22,1/(,10) :: 22, 1 (,1):>: 2, fl (; 1) = 21, ndex Tenns-Quaternary linear cdes, bunds n minimum length,. NTRODUCTON Let GF( 1]) dente the Galis field f g elements, and let F (n, q) dente the vectr space f all rdered 11 -tuples ver GF( q), A linear cde C f length 11 and dimensin k ver GF( J.) is a k-dimensinal subspace f Hn, g). Such a cde is called an [1, :, d: q] -cde if its minimum Hamming distance is d. A central prblem in cding thery is that f ptimizing ne f the parameters, k, and d fr given values f the ther tw. Tw equivalent versin are: Prblem 1: Find dq(n, k), the largest value f d fr which there exists an [11, k, d; g]-cde. Prublem 2: Find n q (k, d), the smallest value f n fr which there exists an [n, /"d;g]-cde. A cde which achieves ne f these tw values is called ptimal Manuscript received September 9, 199; revised May, 199. The authrs are with the Department f Mathematics, Technical University, 00 Gabrv, Bulgaria. EEE Lg Number /9$ EEE