SOME CONSTRUCTIONS OF OPTIMAL BINARY LINEAR UNEQUAL ERROR PROTECTION CODES

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1 Philips J. Res. 39, ,1984 R 1097 SOME CONSTRUCTIONS OF OPTIMAL BINARY LINEAR UNEQUAL ERROR PROTECTION CODES by W. J. VAN OILS Philips Research Labratries, 5600 JA Eindhven, The Netherlands Abstract This paper describes a number f cnstructins f binary Linear Unequal Errr Prteetin (LUEP) cdes. The separatin vectrs f the cnstructed cdes include thse f all ptimal binary LUEP cdes f length less than r equal t 15. AMS: 94BOS, 94B Intrduetin Cnsider a binary linear cde C f length n and dimensin k with generatr matrix G t be used n a binary symmetric channel. In many applicatins it is necessary t prvide different prteetin levels fr different cmpnents mi f the input message wrd m. Fr example in transmitting numerical (binary) data, errrs in the mre significant bits are mre serius than are errrs in the less significant bits, and therefre mre significant bits shuld have mre prtectin than less significant bits. A suitable measure fr these prteetin levels fr separate psitins in input message wrds is the separatin vectr 1). Definitin Fr a binary linear [n,k] cde C the separatin vectr s(g) = (s(gh, s(gh,..., s(g)k) with respect t a generatr matrix G f C is defined by S(G)i:= min (wt(m G)I m E (0, IJk, mi = IJ, where wt(.) dentes the Hamming weight functin. This separatin vectr s(g) guarantees the crrect interpretatin f the i th message bit whenever Nearest Neighbur Decding (ref. 2 p. 1) is applied and n mre than (S(G)i - 1)/2 errrs have ccurred in the transmitted cdewrd 1). A linear cde that has a generatr matrix G such that the cmpnents f the crrespnding separatin vectr s(g) are nt mutually equal is called a Linear Phillps Jurnl f Research Vl. 39 N

2 W. J. van Oils Unequal Errr Prteetin (LUEP) cde 1). By permuting the rws f a genneratr matrix 0 we may btain a generatr matrix GJ' fr the cde such that s(0 ') is nnincreasing, i.e. s(0 ');~ s(0 ');+1 fr i = 1,2,..., k - 1. In this paper we always assume that the rws in generatr matrices are s rdered that the crrespnding separatin vectrs are nnincreasing. Any LUEP cde C has a s-called ptimal generatr matrix 0*. This means that the separatin vectr s(o*) is cmpnentwise larger than r equal t the separatin vectr s(o) f any generatr matrix 0 f Cl), dented by s(o*) ~ s(o) (x ~ y means x; ~ y; fr all i). The vectr s = s(o*) is called the separatin vectr f the linear cde C. We use the ntatin [n,k,s] fr C. Fr any keln and seink we define n(s) t be the length f the shrtest binary linear cde f dimensin k with a separatin vectr f at least s, and nex(s) t be the length f the shrtest binary linear cde f dimensin k with separatin vectr (exactly) S3,4). An [n(s),k,s] cde is called length-ptimal"). It is called ptimal if an [n(s),k,t] cde with t ~ s, t =F s des nt exist ê'). In refs 3 and 4 a number f bunds fr the functins n(s) and nex(s) are derived. In ref. 5 methds fr cnstructing LUEP cdes frm shrt her cdes are described. In refs 3 and 4 an incmplete list f the separatin vectrs f the ptimal binary LUEP cdes f length less than r equal t 15 is given. In this paper we prvide the cmplete list f the separatin vectrs f all ptimal binary LUEP cdes f length less than r equal t 15, tgether with examples f generatr matrices having these separatin vectrs. Furthermre, we give a number f cnstructins f infinite series ptimal binary LUEP cdes. 2. Cnstructins Table I prvides the separatin vectrs f all ptimal binary LUEP cdes f length less than r equal t 15.In this table, n dentes the length f the cde, k dentes the dimensin, and d(n, k) dentes the maximal minimum distance f a binary cde f length n and dimensin k. The brackets and cmmas cmmnly appearing in separatin vectrs have been deleted. Only in the cases where a cmpnent f a separatin vectr is larger than 9, it is fllwed by a pint (.). Examples f cdes having the parameters given in table I are cnstructed belw. The bunds in ref. 4 can be used t shw that certain LUEP cdes are ptimal. They are als useful in shwing that table I is cmplete. In cases where these bunds did nt wrk, methds f exhaustive search were used t shw that cdes with certain parameters d nt exist. Table I is the same table as table I in ref. 4, extended by the parameters [14,10,( )], [15,3,(994)], [15,8,( )], [15,8,( )], [15,8,( )] and [15,11,( )].In (ref. 4 table I) n references t cnstructins were given, which has been dne in this paper. 294 Phllips Jurnalf Research Vl.39 N

3 Sme cnstructins f ptimal binary linear unequal errr prtectin cdes TABLE The separatin vectr f all binary ptimal LUEP cdes f length less than r equal t 15. n k d(n,k) separatin vectr A A A A A A A 62, A A A A 72, A 622, C A A 42222, J A A 82, A 722, C 644, G A 6222, C A 52222, J 44442, B A , J A A 92,184, A 822, C 744, L A 7222, C 6444, G A 62222, C A , J , J A , J A A 10.2, 194, A 922, C 844, x A 8222, C 7444, E A 72222, C 64444, G A , J , B A , J , J A , J A A 11.2, 110.4, A 10.22, C 944, E 864, K2 774, te, A 9222, C 8444, s, A 82222, C 74444, E 66444, M A , C , G A , J , J I Phllips Jurnlf Research Vl. 39 N

4 W. J. van Oils n k d(n,k) separatin vectr TABLE I (cnt.) 12' 8 3 A , J , J A , J A A 12.2,111.4, 110.6, A 11.22, C 10.44, «, 964, E 884, L A , C 9444, L 8644, F 7744, x, A 92222, C 84444, x, 76444, L 66664, H A , C , D , M A , J , B , J , x, A , J , J A , J , J A , J , A A 13.2, 112.4,111.6, I A 12.22, C 11.44, L 10.64, s, 984, x, A , C , KI 9644, L 8844, L A , C 94444, L 86444, F 77444, N A , C , E , L , J A , C , J , Q , M A , J , J , J , K l A , J , J A , J , J A , J A A 14.2, 113.4, 112.6, A 13.22, C 12.44, te, 11.64, x, 10.84, L 10.66, K2 994, «, A , C , L , x, 9844, x, A , C , x, 96444, L 88444, L A , C , L , I_< , J , «, , A , C , P , L , J A , J , B , J , J , L , R , S A , J , J , J , s, A , J , J A , J , J A , :J A Philips Jurnalf Research Vl.39 N

5 Sme cnstructins f ptimal binary linear unequal errr prtectin cdes In this paper we frequently use tw results f ref. 4. Hence we repeat these results in the fllwing tw therems. Therem 1 (ref. 4,.therem 12) Fr any k e IN and nnincreasing se INk we have that hlds fr any ie {I,..., kj, where A _ {Sj - ls)2j fr j < i SJ.- rsj/21 fr i> i, (where l=l dentes the largest integer smaller than r equal t x, and [x] dentes the smallest integer larger than r equal t x). Therem 2 (ref. 4, crllary 14) Fr any kein and any nnincreasing sein\ fllwing inequalities, a. n(sl S2,..., Sk) ~ SI + n(fs2/21,..., rsk/2l), k b. n(sl S2,..., Sk) ~ L rs;/2 i - 1 l- i=1 the functin n(s) satisfies the Cnstructin A Fr n,kein, n ~ k + 1, the k by n matrix (I 0,_"._,_, (1) is a generatr matrix f an ptimal binary [n,k,(n - k + 1,2,2,...,2)] cde (h dentes the identity matrix f rder k,ok-l,n-k-l dentes the all-zer k - 1. by n - k -1 matrix). Prf It is easy t check that the parameters f the cde are crrect. Furthermre by therem 2b the length f a k-dimensinal binary cde with separatin vectr (n - k + 1,2,2,...,2) is at least n, and with separatin vectr larger than (n - k + 1,2,2,...,2) is at least n + 1.(by s» t (s larger than t) we mean s ~ t, s =1= t). Phillps Jurnal f Research Vl. 39 N

6 W. J. van Oils Cnstructin B Fr kein, k~ 4, the k by 2k - 1 matrix h-l 1 (2) is a generatr matrix f an ptimal binary [2k - l,k,(k - 1,3,3,...,3)] cde. Prf It is easy t verify that the parameters f the cde are crrect. By therem 2b, we have that the length f a k-dimensinal binary cde with separatin vectr (k - 1,3, 3,..., 3) is at least 2k - 1. Applicatin f therem 2b t a k-vectr S with SI~ k and Si ~ 3 fr i = 2,..., k shws that n(s) ~ 2k. Applicatin f the therems 1 and 2 t a k-vectr s such that SI = k - 1, S2 ~ 4, Si ~ 3 fr i = 3,..., k - 1, and Sk = 3 shws that nex(s) ~ 3 + nts, - 1,..., Sk-l - 1) ~ 3 + SI n(f(s2-1)/21,..., r(sk-l - 1)/21) ~ 3 + k n(2, 1, 1,..., 1) k-2 ~ 3 + k k - 1 = 2k. Furthermre it is nt difficult t check that a binary [2k - 1, k, (k - 1,4,4,...,4)] cde des nt exist. Finally, by therem 2b the length f a k-dimensinal binary cde with a separatin vectr f at least (k - 1, 5,4,4,...,4) is at least 2k. These bservatins shw that the cde in cnstructin B is ptimal. Cnstructin C Fr n,ke IN, n ~ max{2k,k 1 + 4}, the k by n matrix h-l (3) 11 is a generatr matrix f an ptimal binary [n,k,(n - k, 4, 4,...,4)] cde. 298 Phillps Jurnalf Research Vl.39 N

7 Sme cnstructins f ptimal binary linear unequal errr prtectin cdes Prf Similar t the prf f cnstructin A. Cnstructin D Fr p, q e IN, p ~ q ~ 2, the p + q + 2 by 2p + 3q + 3 matrix q-l i-, lp t, (4) is a generatr matrix f an ptimal binary [2p + 3q + 3,p + q + 2,(p + q + 2, 2q + 2,4,4,...,4)] cde. Prf Similar t the prf f cnstructin A. Cnstructin E Fr p,q,rein, p~3, r~2, q~r-p+2, the p by (2p+q+2r-4) matrix Ip-2 Ip-2 1 (5) ~ r r q is a generatr matrix f an ptimal binary [2p + q + 2r - 4,p,(p + q + r - 2, 2r, 4, 4,...,4)] cde. Phllips Jumal f Research Vl. 39 N

8 W. J. van Gils Prf Similar t the prf f cnstructin A. Cnstructin F Fr p,qeln, p~ 3, 2, «>» - 2, thep by p + 3q matrix Ip-2 Ip : 1 1 (6) q + 1 q + 1 q - (p -2) is a generatr matrix f an ptimal binary [p + 3q,p,(2q + 1, 2q +.1,4,4,...,4)] cde. Prf Similar t the prf f cnstructin A. Cnstructin G Fr peln, the 2p by 4p matrix Ip p Ip (7) is a generatr matrix f a binary [4p,2p,(p + 2,p + 2,4,4,...,4)] cde. Fr p = 2, 3 the cdes are ptimal, but in general they are nt. In ref. 6 the cdes frm cnstructin G are treated extensively, the weight enumeratrs and autmrphism grups are determined cmpletely and a majrity lgic decding methd fr these cdes is given. Fr p = 3 we btain a [12,6,(5,5,4,4,4,4)] ptimal LUEP cde. By deleting the rw and clumn pairs (6,4), (5,3) and (4,2) successively we btain [11,5,(5,5,4,4,4)], [10,4,(5,5,4,4)] and [9,3,(5,5,4)] ptimal LUEP cdes respectively. 300 Phllips Jurnalf Research Vl.39 N

9 Sme cnstructins f ptimal binary linear unequal errr prtectin cdes Cnstructin H Fr pein,»» 3, the (p + 2) by (4p + 1) matrix (8) is a generatr matrix f a length-ptimal LUEP cde. binary [4p + l,p + 2,(2p,2p,5,5,..., 5)] Prf It is easy t check that the cde has the given parameters. By therem 2b the length f a (p + 2)-dimensinal binary cde with separatin vectr (2p,2p,5,5,...,5) is at least 4p + 1. Fr p = 3 this cnstructin gives a [13,5,(6,6,5,5,5)] ptimal LUEP cde. Furthermre table I refers t the fllwing trivial cnstructins. 1 Cnstructin I Fr p,qein, p > q, the 2 by (p + 2q) matrix lj [ ~ p q q (9) is a generatr matrix f an ptimal binary [p + 2q,2,(p + q,2q)] LUEP cde. Cnstructin J If the matrix Ol has separatin vectr S(Ol) such that s(olh -;;::. 2, then the matrix has separatin vectr S(02) = (s(01),2). Ol (10) Philips Jurnalf Research Vl. 39 N

10 W. J. van Gils Cnstructin Ki If the matrix Gl has separatin vectr S(Gl) then the matrix G2 ;= [ Gl lei]' (11) where ei is the vectr with a 1 n the i th psitin and zers elsewhere, has separatin vectr S(G2) = s(gl) + ei.. The fllwing therem can be used t determine whether cnstructin K, gives an ptimal cde.. Therem 3 If s is such that fr all t ~ s, t =1= s, it hlds that net) > nes) and if G is a generatr matrix f a binary ptimal [r + n(s),k,(r,2s)] cde, then the cde generated by [G I ell ell... 1 el] is a binary ptimal [r + t + n(s),k,(r + t,2s)] +--t----+ cde fr t in IN arbitrary. Prf Let sand that G fulfill the cnditins mentined abve. By therem 2a we have a) n(r + t,2s) ~ r + t + n(s). b) n(r + t + 1,2s) ~ r + t nes) > r + t + nes). c) n(r+ t,2s + u) ~ r+ t + nasl + ud21,...,[sk-l + uk-d2l) ~ r+ t nes) fr u ~ 0, u =1=. Cmbinatin fa), b) and c) shws that the cdegenerated by [G I ell ell.. 1 el] is ptimal. +--t----+ Cnstructin L Adding an verall parity-check bit t a binary [n,k,s = (SI,..., Sk)] cde gives a binary [n + 1,k,s' = (2l(Sl + 1)/2J,..., 2l(Sk + 1)/2J)] cde. Spradic cnstructins referred t in table I are the fllwing. Cnstructin M The 7 by 14 matrix (12) 302 Philip, Jurnalf Research Vl.39 N

11 ! Sme cnstructins f ptimal binary linear unequal errr prtectin cdes is a generatr matrix f an ptimal binary [14,7,(5,5,5,5,4,4,4)] LUEP cde. Deleting the first clumn and the last rw frm the matrix in (12) gives an ptimal binary [13,6,(5,5,5,5,4,4)] cde. Deleting the first tw clumns and the last tw rws frm the matrix in (12) gives an ptimal binary [12,5,(5,5,5,5,4)] LUEP cde. Cnstructin N Applicatin f [5,cnstructin 1] with m = 1, q = 2 and G1 a generatr matrix f the [7,4,(3,3,3,3)] Hamming cde gives an ptimal binary [14,5,(7,6,6,6,6)] LUEP cde. Cnstructin 0 The 6 by 15 matrix (13) is a generatr matrix f an ptimal binary [15,6,(7,6,5,5,5,4)] LUEP cde. Cnstructin P The 7 by 15 matrix (14) is a generatr matrix f an ptimal binary [15,7,(7,6,4,4,4,4,4)] Cnstructin Q By deleting the 8 th clumn frm the matrix in (14) we btain LUEP cde. a generatr matrix f an ptimal binary [14,7,(6,5,4,4,4,4,4)] cde. I Cnstructin R. The 8 by 15.matrix Philips Jurnalf Research Vl.39 N

12 W. J..van Gils G (15) where G is the matrix in (12), is a generatr [15,8,(5,5,5,5,4.,4,4,3)] LUEP cde. matrix f an ptimal binary Cnstructin S The 8 by 15 matrix (16) is a generatr matrix f an ptimal binary [15,8,(5,5,5,4,4,4,4,4)] LUEP cde. REFERENCES 1) L. A. Dunning and W. E. Rbbins, Infrm. Cntr. 37,150 (1978). 2) F. J. MacWilliams and N. J. A. Slane, The thery f errr-crrecting cdes, Nrth- Hlland Mathematical Library, Amsterdam, Vl. 16, ) W. J. van Gils, EUT-Rep. 82-WSK-02, Department f Mathematics and Cmputing Science, Eindhven University f Technlgy, Eindhven, The Netherlands, June ) W. J. van Gils, IEEE Trans. Infrm. Thery IT-29, 866 (1983). 6) W. J. van Gils, IEEE Trans. Infrm. Thery IT-30, 544 (1984). 6) L. M. H. E. Driessen, IEEE Trans. Infrm. Thery IT-30, 392 (1984). 304 Phllips JurnnI f Research Vl.39 N