AN APPLICATION OF INCOMPLETE BLOCK DESIGNS. K. J. C. Smith University of North Carolina. Institute of Statistics Mimeo Series No. 587.

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1 AN APPLICATION OF INCOMPLETE BLOCK DESIGNS TO THE CONSTRUCTION OF ERROR-CORRECTING CODES by K. J. C. Sith University of North Carolina Institute of Statistics Mieo Series No. 587 August, 1968 This research was supported by the National Science Foundation Grants No. GP-5790 and No. GU-2059 and by the Ary Research Office, Durha, Grant No. DA-ARO-D-3l-l24-G9l0. DEPARTMENT OF STATISTICS UNIVERSITY OF NORTH CAROLINA Chapel Hill, N. C.

2 1. Introduction Historically, balanced and partially balanced incoplete block designs have been used priarily for statistical applications in the design of experients. Recently, incoplete block designs have been used by several authors in constructing error-correcting codes. This approach has yielded useful and efficient constructions of codes. This paper presents an application of incoplete block designs to the construction of error-correcting codes which ay be decoded using a relatively siple ajority logic decoding procedure. An identity atrix, I, is adjoined to the incidence atrix, N, of a balanced or partially balanced incoplete block design. The resulting atrix is taken as the parity check atrix of a linear code; a relatively siple ajority logic decoding procedure for error correction ay be used for the code. A brief introduction to incoplete block designs and to linear codes is given in sections 2 and 3 for the reader unfailiar with either area.

3 2 2. Incoplete Block Designs 2.1 Balanced Incoplete Block Designs A balanced incoplete block (BIB) design is an arrangeent of v objects, usually referred to as treatents, into b sets, called blocks, such that the following conditions are satisfied: (i) (ii) each block contains k*l distinct treatents; each treatent occurs in r*l distinct blocks; blocks. (iii) each pair of treatents occur together in exactly A different The variables v, b, r*, k*, A are referred to as the paraeters of the design. The existence and construction of BIB designs are treated in detail by Bose (1939), Bose (1969), Hall (1967), and Ryser (1963), aong others. Ryser (1963) uses the ter (v,b,r*,k*,a)- configuration for a BIB design with paraeters v,b,r*,k*,a. The five paraeters of a BIB design satisfy the following relations (see Bose (1969), for exaple): (2.1.1) vr* = bk*, (2.1.2) A(v-l)= r*(k*-l). Obviously, A~ r*. If O<A<r*, then Fisher's inequality, Bose (1969), states that (2.1. 3) b~ v. If b = v, then, fro equation (2.1.1), r* = k*. In this case, the BIB design is said to be syetrical. ITraditionally, k and r are used as paraeters of a design. We shall use k* and r* to avoid confusion with notation used in coding theory.

4 3 2.2 Partially Balanced Incoplete Block Designs Before proceeding, we need the concept of an association schee, Bose and Shiaoto (1952). An -class association schee is a relation between v treatents satisfying the following conditions: ( a) th any two treatents are either 1st, 2nd,..., or ' associates, the relation of association being syetrical; ( b) h t t t h.th. h d d eac rea en as n i 1 assoc1ates, t e nuber n i being in epen ent of the treatent chosen. ().f d Q th. t th th b f t t t h' h c 1 ~ an ~ are 1 assoc1a es, en e nu er 0 rea en s w 1C are jth associates of ~ and k th associates of ~ is P;k and is independent of t he pari 0 f i th assoc1ates. d ~ an ~. Q (i, j, k=l,2,.., ) i i Pjk = Pkj (1939). i The paraeters v, n i, Pjk are not all independent. For exaple, Other relations aong the paraeters are given by Bose and Nair Given an -class association schee of v treatents, a partially balanced incoplete block (PBIB) design is an arrangeent of the v treatents into b blocks such that (i) (ii) (iii) each block contains k* distinct treatents; each treatent occurs in r* distinct treatents; each pair of treatents which are i th associates occur together in exactly A. 1 blocks (i=1,2,..,). For a PBIB design based on an -class association schee with paraeters v, n., P:k' it is shown by Bose and Nair (1939), for exaple, that 1 J (2.2.1) vr* = bk* and

5 4 (2.2.2) In general, the nuber of blocks need not be as large as the nuber of treatents. A BIB design with paraeters v,b,r*,k*,a ay be considered a PBIB design based on an association schee with one class. 2.3 Incidence Matrix ~~ Design design as Let us arbitrarily label and order the treatents of a PBIB (or BIB) and the blocks, siilarily, as B l ' B 2,, B b We shall say that V. is incident with B. if the treatent V. occurs in the J 1 J block B.. ( i=l, 2,.., b i j =1, 2,..., v.) 1 (2.3.1) The b X v incidence atrix, N, of the design is defined as the atrix where (2.3.2) 1, if Vj ts incidentwith (occurs in) Bii 0, otherwise. (i=1,2, 00.,bi j=1,2,. 00'.) Clearly, each row of N contains k* 1 's and colun of N contains r* 1 'so

6 5 3. Linear Error-Correcting Codes 3.1 Introduction We state here a few basic concepts of linear error-correcting codes. The reader is referred to Peterson (1963) or to Berlekap (1968) for further details. Only binary codes will be considered here. A nuber of essages are to be transitted over a noisy channel fro a source to a destination. Each essage corresponds to a sequence of n binary sybols, say 0 and 1, called a codeword. The n sybols of a codeword t,2 = (t,t,...,t ) are transitted consecutively over the channel. Rando l 2 n "noise" in the channel ay cause a 0 to be received as a 1 or vice versa. If this happens, we shall say that a transission error has occurred. The corresponding received sequence, ~' = (r l,r 2,...,r n ) is not necessarily a codeword. Suppose~' =!' + ~', where~' = (e l,e 2,...,e n ) is the unknown error vector; each coordinate of e' is a 0 or a 1 and the addition is in binary, i.e. over GF(2). The nuber of nonzero coordinates of e' is the nuber of errors which have occurred in transitting!'. We say that it is possible to correct s errors if, assuing at ost s errors have occurred in transitting a codeword, it is possible to deterine correctly which coordinates of the corresponding error vector are nonzero. The correct transitted codeword ay then be obtained by copleenting these coordinates of the received vector. This procedure is called decoding. A binary linear (n,k) code C is a k-diensional subspace of the n diensional vector space over GF(2). k Each of the 2-1 nonzero vectors of C 2The notation a' = (a,a,..,a ) l 2 n colun vector. denotes a row vector; a denotes a

7 6 is a codeword; k is the nuber of inforation sybols of the code C. The redundancy of C is r = n-k. A atrix whose rows span C is a generator atrix of the code. The orthogonal copleent of C, in the usual algebraic sense, is the ~ ~ C'. A parity check atrix of C is a atrix whose rows span the dual code C'. If H is a parity check atrix of C, then the vector~' = (gl,g2'.,gn) is a codeword of C if and only if (3.1.1) Each equation of (3.1.1), corresponding to the rows of H, is a parity check equation. Suppose t' is a transitted codeword and r' is the corresponding received vector. and The error vector is e' Ht = 0 = r' t ' -. Since t' is a codeword of C, then (3.1. 2) Hr = Ht+He = He =!o' say. The vector s = Hr is called the sydroe of the received vector r'. Equation (3.1.2) ay be used for decoding. The syndroe of a received word is ca1- culated and the equation He = s ay be solved, in soe instances, for the unknown error vector ~', assuing at ost, say, of the coordinates of e' are nonzero. 3.2 Systeatic codes If G is the generator atrix of a linear code C and G* is obtained fro G by colun perutations, then G* generates a linear code c* defined to be equivalent to C. Given a (n,k) linear code C, we can find an equivalent code C* for which the generating atrix is

8 7 where r k is the k x k identity atrix and P is a k x (n-k) atrix. Every codeword of C* is a linear cobination of the rows of G* and is of the for (c I ' c 2 '.., c k ' c I PII+c 2 P21+.+c~kl '..., c I Plr+c 2 P 2r '. '+'k J:k r), where c l,c 2,...,c k are arbitrary and P = (Pij)kxr' r = n-k. Thus the first k coordinates of a codeword of c* ay be arbitrarily chosen. The reaining coordinates are linear cobinations of these coordinates. The first k coordinates are inforation places and the reaining r coordinates are redundant places. Such a code is called a systeatic ~. We shall now consider a systeatic (n,k) code C with generator atrix for soe P. Let T H = [p, I J, r where p T denotes the transpose of P. Over GF(2), CpT I r ] G~] = = pt + pt o Thus the vector space generated by the rows of H is orthogonal to that generated by G, i.e., H is a parity check atrix for C. Since the first k coordinates of a codeword in a systeatic code copletely specify the codeword, only these sybols in a received vector need be decoded. The reaining r redundant sybols ay be deterined fro the first k sybols to for the decoded codeword if necessary.

9 8 3.3 Majority-logic decoding The technique of ajority-logic decoding of linear codes was developed by Massey (1963) for particular linear codes. This procedure is based on the concept of a set of parity check equations orthogonal on a given sybol. Let C be an (n,k) linear code and suppose it is possible to find a set of J vectors of the dual code C', say (3.3.1) hj = (hjl,hj2,,hjn)' j = 1,2,...,J such that for fixed, = 1,,n h. = 1 for each j = 1,2,...,J J (3.3.2) = 0 for all but at ost one j = 1,2,...,J and for any fixed u F:. of these J That is, each vectors of the J vectors h~ has th coordinate 1 and at ost one -J has u th coordinate 1 for any u F:. Then the set of J equations (3.3.3) h'lel+h'2e2+...+hj e =s., u=1,2,.,j J J,n n J are said to be orthogonal ~ the sybol!ro' (Massey (1963». If the set of equations (3.3.3) corresponds to a set of parity check equations for a linear code C, then a ajority-logic procedure ay be used to deterine the error sybol e and thus decode the received sybol r into the corresponding sybol t fro the following theore. Theore Suppose a set of J parity check equations (3.3.3) orthogonal as the sybol e ay be found for a linear code C. Then, provided at ost ~f2j3 errors have occured, the sybol e is given correctly by the following rule: 3The notation [x] denotes the greatest integer less than or equal to x.

10 (1) e is that value of GF(2) which is assued by the ajority (greatest fraction) of the [s.}, provided such a value exists; J (2) e is zero if the [s.} take on the values 0 and 1 with equal frequency. J This theore is proved by Massey (1963). We shall oit the proof here 9 and prove a generalization of the theore which does not assue the orthogonality of the parity check equations on a particular sybol. This generalization appears in Rudolph (1967) and Sith (1967). Theore Let C be a linear (n,h) code and let (3.3.4) h.!el+h '2e2+".+h j = 1,2,...,J 0 e = So J J In n J be a set of J parity check equations such that (3.3.5) hj = 1,j = 1,2,...,J. Assue that for any u ~, (3.3.6) h = 1 for at ost A subscripts j = 1,2,..,J. ju Then, provided at ost [J/ 2A ] errors have occurred, the sybol e is given correctly by the following rule: (1) the (2) e is that value of GF(2) which is assued by the greatest fraction of [s.}, if such a ost frequent value exists. J e is zero if the [so} take on the values 0 and 1 with equal frequency. J Proof: Suppose at ost t = [J/ 2A ] errors have occurred. Then at ost t of the sybols e l,e 2,..,e n are nonzero. If all sybols other than e were zero, then (3.3.7) s =e,j=1,2,...,j j and the decision rule of the theore is correct. If one of the other sybols is nonzero, then at ost A of the s. in equation (3.3.7) are different fro J e. In general, if x other sybols are nonzero, at ost XA of the So in J equation (3.3.7) are different fro e ' since each of these sybols can

11 affect at ost A of the s. J If ern = 0, then if t or fewer of the error sybols are nonzero, at ost ta of the s. are nonzero. Since J~2tA, then the decision rule of the theore J gives the correct value of e. If e = 1, then if (t-l) or fewer of the other error sybols are nonzero, at ost (t-l)a of the s. are different fro e. Since J >2(t-l)A, then the J greatest fraction of the s. are equal to 1 and the decision rule of the J theore gives the correct value of e. 3.4 Majority Decodable Codes A nuber of linear codes have been shown to be decodable using ajority 10 logic techniques. Aong these are the Reed-Muller codes and their generalizations discussed by Kasarni et al (1968) and Weldon (1968), as well as the codes tered "geoetric codes" by Goethals and Delsarte (1967), Rudolph (1967) and Sith (1967). Townsend and Weldon (1967) have investigated a class of codes called quasi-cyclic self-orthogonal codes, to which a ajority logic decoding algorith ay be applied. With the exception of the latter codes and soe special cases of the other codes entioned above, the proble of deterining a general expression for the nuber of inforation sybols in these codes is difficult and has not yet been copletely solved. To avoid this difficulty, we shall present in section 4 a class of linear codes which ay be ajority logic decodable and for which the nuber of inforation sybols is iediately deterined. Special cases of these codes are equivalent to the quasi-cyclic self orthogonal codes discussed by Townsend and Weldon (1967).

12 11 4. Systeatic ~ Design Codes 4.1 Definition We define a large class of linear codes, which we shall call systeatic block design codes, as follows. Let N be the b x v incidence atrix, defined in section 2.3, of a balanced or partially balanced incoplete block design D with paraeters v,b,r*,k*,al,...,a. Let I be an identity atrix of order b. The systeatic b block design code (associated with the design D) is the (binary) linear code orthogonal to the row space of the atrix (4.1.1) that is, the linear code for which H is a parity check atrix. The systeatic block design code associated with the atrix H has the following iediate properties. (1) The length of the code is n = v+b (2) (3) (4) (5) (4.1.2) (6) (4.1. 3) The redundancy is r = b. The nuber of inforation sybols is k = n-r=v. The code is systeatic but not necessarily cyclic. The generator atrix of the code is T G=[I,N] v The inforation rate R =kin is v 1 R = V+b = l+b/ v Clearly, Rs1 if and only if b~v. For a BIB design b~v. Thus we ay state Lea The inforation rate of a systeatic block design code associated With a BIB design is at ost 1. In searching for a high rate (R>l, say) systeatic block design code, we

13 ust therefore restrict our attention to codes associated with PBIB designs for which b<v. Soe exaples of such codes will be given in section Encoding ~ Decoding Since the codes are systeatic, the encoding or construction of codewords is relatively siple. The first k = v sybols ay be chosen arbitrarily; the reaining r = b redundant sybols are linear cobinations, deterined by the generator atrix in equation (4.1.2), of the k inforation sybols. A relatively siple ajority logic decoding procedure ay be used for the systeatic block design codes. We state this as a theore. Theore Let C be a systeatic block design code associated with a (partially) balanced incoplete block design D with paraeters v,b,r*,k*, (4.2.1) Up to [r*/ 2A J errors ay be corrected using a one-step ajority logic decoding procedure. Proof: It is necessary to correct only the inforation sybols, for the redundant sybols ay then be deterined fro the encoding procedure. EaCh row of the parity check atrix of the code deterines a parity check equation. If N is the incidence atrix of the associated design, then the parity check atrix for the code is H = [N, I]. Each of the first k = v inforation sybols corresponds to a treatent in the design. Each row of H corresponds to a block. The first k entries in a row are 1 or 0, according to whether the appropriate treatent occurs or does not occur in the corresponding block. Each treatent occurs in r* blocks.

14 and the r* rows of H cor Consider anyone of the v treatents, say V a responding to the blocks in which V occurs. Each of the r* parity check a equations deterined by these rows, (4.2.2) h e + h e + jl l j2 2 say 13 Let us relabel the rows such that these are the first r* rows of H. Then for j = 1,2,...,r*, = 1. If treatent V and V are i th associates, then u a Moreover, h. = 1 for at ost Xi subscripts j = 1,2,..,r* JU and for any fixed u p a, l~u~v. h ja = 1 for at ost one j = 1,2,,r* if v+l~a$n. Thus, if X = ax Xi' then l$i$ro (4.2.3) fixed u p a. h. = 1 for j = 1,2,...,r* Ja h ju = 1 for at ost X subscripts j = 1,2,...,r* and for any Hence, fro Theore 3.3.2, a ajority logic decoding procedure ay be used to deterine e. This will a have occurred. The procedure is give e a correctly if at ost [r*/ 2X ] to consider the set of r* parity check errors equations corresponding to the blocks in which treatent V occurs. The a sybol e a is given as that value of GF(2) which is assued by the greatest fraction of the corresponding s. in the syndroe vector. J fraction exists, then e a is zero. If no such greatest This procedure is repeated for each of the reaining v-i inforation sybols and yields the inforation sybols of the codeword corresponding to a received word correctly, provided at ost []tjerrors have occurred.

15 Systeatic Block Design Codes Associated ~~ Designs A systeatic block design code associated with a BIB design with paraeters v,b,r*,k*,a is a (n,k) linear code with n = v+b and k = v. Th~ ajority-logic decoding procedure described in section 4.2 will correct up to [r*/2aj errors. In this section, we give a few exaples of these codes associated with particular BIB designs. The literature on BIB designs is extensive and includes any constructions for a wide range of design paraeters. The reader is referred to Bose (1969) for a coprehensive treatent of such designs and to ethods of constructing the designs entioned below. BIB designswith k* = 3 and A = 1 are called Steiner triple systes. It ay be shown that these conditions iply that r* ust be of the for 3t+l or 3t for soe positive integer t. In the case r* = 3t+l, the paraeters are v = 6t+3, b = (3t+l)(2t+l), r* = 3t+l, k* = 3, A =1. This series of designs is referred to as the T l series and exists for all values of t. The systeatic block codes associated with the T l designs have n = v+b = (2t+l) (3t+4) k = v = 6t+3 3 R = kin = 3t+4 and the ajority decoding procedure described earlier will correct up to [r*/2j = [3~+lJ errors. Table lists these paraeters of the codes with soe of the T l series of designs.

16 15 TABLE Systeatic Block Design Codes Associated With T l Series v = 6t+3, b = (3t+l)(2t+l), r* = 3t+l, k* = 3, A = 1 t Code Inforation Errors Rate Length Sybols Corrected These codes have low rates and only a relatively sall nuber of errors are guaranteed correctable by the ajority decoding procedure. A different series of BIB designs with ~ = 1 are the BIB designs constructed fro finite projective planes. These designs are often called the orthogonal series 2 or OS2 series. The paraeters are 2 v = b = s +s+l, r* = k* = s+l, ~ = 1. Such designs are known to exist when s is a prie power. The associated systeatic block design codes have n = v+b k = v = 2 = 2(s +s+l), 2 2( s +5+1), R = k/ =!, n and up to ~*/ 2J = [s;lj errors ay be corrected by our decoding procedure. These paraeters of soe codes are exhibited in Table

17 16 TABLE Systeatic Block Design Codes Associated With OS2 Series 2 v = b = s +s+l, r* = k* = s+l, A = 1 s Code Inforation Errors Rate Length Sybols Corrected While these codes have rate.5, the nuber of errors corrected is still relatively sall. 4.4 Systeatic Block Design Codes Associated with ~ Designs A systeatic block design code associated with a PBIB design with paraeters v, b, r*, k *, A. l ',A. has [r*/2aj errors ay be corrected described in Section 4.2. n = v+b and k = v. If A. = ax A.., then up to. 1 1 S1: using the ajority-logic decoding procedure Of particular interest are the PBIB designs with b<v, for the rates of the associated codes are greater than 1. We shall consider here codes associated with soe PBIB designs based on partial geoetries, Bose (1963), for which b<v. If the roles of treatents and blocks of a BIB design with A. = 1 are interchanged, then the resulting design, called the dual design, is a PBIB

18 17 design with paraeters b,v,k*,r*,l,o. This notation iplies that the nuber of treatents is b, etc. Consider the T series of BIB designs, discussed in section 4.3. The l dual designs are PBIB designs with paraeters v = (3t+l)(2t+l), b = 6t+3, r* = 3, k* = 3t+l, Al = 1, A 2 = 0. The resulting systeatic block design codes have n = (2t+l) (3t+4) k = (3t+l) (2t+l) R = 3t+l 3t+4 and the ajority decoding procedure will Table exhibits these properties of correct up to [3/~ = 1 error. soe such codes. TABLE Systeatic Block Design Codes Associated With Duals of T 1 Designs v = (3t+l)(2t+l), b = 6t+l, r* = 3, k* = 3t+l, Al = 1, A 2 = t Code Inforation Errors Rate Length Sybols Corrected These codes have high rates, although only a single error is guaranteed correctable by the ajority decoding procedure. The latter property is not necessarily a disadvantage, however.

19 A ore interesting PBIB design is that obtained fro the configuration of points and generators on an elliptic non-degenerate quadric Q 5 in the 18 finite projective space PG(5,s). Taking treatents as generators and blocks as points, we obtain a PBIB design with paraeters. 233 v = (s +l)(s +1), b = (s+l)(s +1) r* = s+l, k* This design is discussed by Ray-Chaudhuri (1962) and Bose (1963). Table gives the paraeters of the associated syetric block design code for a few values of s, s being a prie power. TABLE Systeatic Block Design Codes Associated With The PBIB Design v = (s +l)(s +1), b = (s+l)(s +1), r* = s+l, k* = s +1, A 1 =1, A 2 =0 s Code Inforation Errors Rate Length Sybols Corrected Discussion A systeatic block design code associated with an incoplete block design with v treatents and b blocks has rate R -...!- If b~v, then R.< v+b If the associated design is a (P)BIB design with paraeters v,b,r*,k*, AI'..,A, then, fro equation (2.1.1), vr* b =- k*

20 19 and R = k* k*+r* = 1 Since the ore interesting codes are those whose rates are bounded away fro zero, designs for which r*/k* is bounded will be ore useful. The nuber of errors corrected by the ajority logic decoding procedure is [r*/2~. Thus, designs with relatively large values of r* and relatively sall values of A = ax Ai will also be of interest. lsi$. Designs with r* relatively large have not yet been investigated as thoroughly as those with r* $.20, Until the advent of coputers, designs with large values of r* and k* have been ipractical for statistical application. However, further research on these designs ay yield iportant results in such areas as the constructions of error-correcting codes. The nuber of errors guaranteed correctable by a ajority decoding procedure of the systeatic block design codes is relatively sall. However, a ore iportant property of a code is the average probability of incorrect decoding. An upper bound on the probability of a decoding error ay be calculated for the systeatic block design codes for which A = 1. This will be the subject of a separate report.

21 20 REFERENCES 1. Berlekap, E. R. (1968). Algebraic Coding Theory, McGraw-Hill, New York. 2. Bose, R. C. (1939). "On the construction of balanced incoplete block designs," Ann. Eugenics, ~, Bose, R. C. (1963). "Strongly regular graphs, partial geoetries and partially balanced designs," ~. :!.. Math., 22, Bose, R. C. (1969). Cobinatorial Probles of Experiental Designs, Vol I, John Wiley and Sons, New York, (to appear). 5. Bose, R. C. and Nair, K. R. (1939). "Partially balanced incoplete block designs," Sankhya,.,!, Bose, R. C. and Shiaoto, T. (1952). "Classification and analysis of partially balanced designs with two associate classes,"!.. ~. Statist. Assoc., ~, Goethals, J. M. and Delsarte, p. (196). "On a class of ajority logic decodable cyclic codes," IEEE Trans. on Inforation Theory, IT-14, Hall, M., Jr. (1967). Cobinatorial Theory, Blaisdell, Wathha, Mass. 9. Kasai, T., S. Lin and W. W. Peterson. (1968). the Reed-Muller codes - Part I: Priitive Inforation Theory, It-14, "New generalizations of codes," ~ Trans. on 10. Massey, J. L. (1963). Threshold Decoding, M.I.T. Press, Cabridge, Mass. 11. Peterson, W. W. (1961). Error-Correcting Codes, M.I.T. Press, Cabridge, Mass. 12. Ray-Chaudhuri, D. K. (1962). "Application of the geoetry of quadrics for constructing PBIB designs," ~. ~. Statist., ~, Rudolph, L. D. (1967). "A class of ajority logic decodable codes," ~ Trans. ~ Inforation Theory, IT-13, Ryser, H. J. (1963). Cobinatorial Matheatics, Carus Math. Monograph No. 14, Matheatical Association of Aerica, Buffalo, N. Y. 15. Sith, K. J. C. (1967). "Majority decodable derived fro finite geoetries," Institute of Statistics Mieo Series No. 561, Departent of Statistics, University of North Carolina.

22 16. Townsend, R. L. and Weldon, E. J., Jr. (1967). "Self-orthogonal quasicyclic codes," ~ Trans. ~ Inforation Theory, IT-13, Weldon, E. J., Jr. (1968). "New generalizations of the Reed-Muller codes - Part II: Nonpriitive codes." IEEE Trans. on Inforation Theory, IT-14,