titvers:tt t>1 NO~~H CAROLINA Depatment of statistics Chapel Hill, N. C. ON A BOUN.D USEFUL IN THE THEORY OF FACTORIAL DESIGNS AND ERROR CORRECTING CODES by R. C. Bose and J. N. Sivastava Apil 1963 Gant No. AF-AFOSR-84-63 Conside a finite pojective space PG(-1,s) of -l dimensions, > 3, based on the Galois field GF, whee s = ph, P being a - s pime. A set of distinct points in o(-l,s) is said to be a non-collinea set, if no thee ae collinea. The m:l.ximum numbe of points in such a non-collinea set is denoted by It is the object of this pape to find a new uppe bound fo ~(,s). ~(,s ) This bound is of impotance in the theoy of factoial designs and eo coecting codes. The exact value of ~(,s) is known when eithe ~ 4 o when s = 2. When ~ 5, s > 3, the best values fo the uppe bound on II}(,s) ae due to Tal1ini,[J.P.7 and Balott! [)}. Ou bound impoves these when s = 3 o "When s is even. Institute of Statistics Mimeo Seies No. 359
ON A BOmID USEFUL IN THE THEORY OF FACTORIAL DESIGNS AND ERROR CORRECTING CODES I by R. C. Bose andj. N. Sivastava Univesity of Noth Caolina = === = == ==== = = = = = = = = == = = = = = = = = = = = = - - = = = = = = = O. Summay Conside a finite pojective space on the Galois field in GF s ' whee PG(-l,s) of -l dimensions, ~ 3, based h s = l', l' being a pime. A set of distinct points PG(-l,s) is said to be a non-collinea set, if no thee ae collinea. 'Fhe'manmum numbe of points in such a non-collinea set is denoted by ~(,s) It is the object of this pape to find a new uppe bound fo ~(,s). This bound is of impotance in the theoy of factoial designs and eo coecting codes. The exact value of ~(,s) is known when eithe < 4 o when s = 2. When ~ 5, s > 3, the best values fo the uppe bound on Il)(,s) ae due to Tallini 12.7 and Balotti ];7. Ou bound impoves these when s = 3 o when s is even. 1. Intoduction R. A. Fishe!!.7, 17 showed that the maximum numbe of factos, which can be accamodated in a symmetical factoial design in which each facto is at s levels and the blocks ae of size s (Whee s is pime), without confounding any main effect o two facto inteaction is (s.l)/(s_l). Bose f~7, genealizing Fishets esult poved the following: Let mt(,s) denote the maximum numbe of points which can be chosen in the finite pojective " ~his eseach was suppoted in pat by the Mathematics Division of the Ai Foce Office of Scientific Reseach unde Gant No. AF-AFOSR-84-63.
~ space pg(-l,s), whee s is a pime o the powe of a pime, so that no t of the points ae dependent. The m t (,s) is the maximum numbe of factos which we can accomodate in a symmetical factoial design in which each facto is at levels and the blocks ae of size 2 s s, so that no t facto o lowe ode inteaction is confounded. Fishes esult follows at once by noting that fo the case t = 2, mt(,s) is simply the numbe of distinct points in PG(-l,s). 1 n Fo a factionally eplicated design ~ x s, consisting of a single block s with s plots o expeimental units, n z:: + k, a slight modification of Bose's agument shows that if it is equied that no d-facto o lowe ode inteaction, should be aliassed with a d-facto o a lowe ode inteaction, then the maximum possible value of n is m 2d (,s). On the othe hand if it is equied that no d-facto o lowe ode inteaction should be aliassed 'With a (d+l)-facto o a lowe ode inteaction, then the maximum value of n would be m 2d + l (,s) The numbe mt(,s) also tuns out to be impotant in infomation theoy. If t hee is an s-ay channel, 1.e. a channel capable of tansmitting s distinct symbols, then fo an (n,k) goup code, with k infomation symbols and fixed edundancy = n - k, the maximum value of n fo which d eos can be coected 'With cetainty is ~d(,s). Similaly the maximum value of n fo which d eos can be coected 'With cetainty and d + 1 eos can be detected is m 2d + l (,s ) This paallelism between the theoy of factional eplications and eo coecting codes has been bought out by Bose 1"L7. Thus the poblem of finding the maximum value of by a constuctive method the coesponding points of m t (,s ), and of obtaining PG(-l,s) is of some impotance. This poblem may be called the packing poblem. Only patial solutions to this poblem ae at pesent know. In the absence of a complete solution a good bound on mt(,s) is desiable. In this pape we shall conside the case t = 3.
Fo this case the packing poblem educes to finding the maximum numbe of points in PG(-l,s) so that no thee ae collinea. Such a set may be called a ~ collinea set, and ~(,s) is then the maximum numbe of points in a non-collinea set in PG(-l,s). Bose E.7 showed that fo the case = 3, (i.e. fo a finite pojective plane) (1.1) (1.2) ~(3,s) = s+l when s is odd, ~(,s) = s+2 when s is even. Fo the case > 3, s = 2, Bose E.7 showed that and the same esult was obtained independently by Hannning l, in connection with binay goup codes coecting one eo and detecting two eos. Fo =4, Bose LE.7 showed that when s is odd (1.4) 2 ~(4,s) = s +1, s > 2 and the same esult was poved to hold tue fo the case when s is even (s > 2) by Qvist 17, the paticula case s=4 having been obtained ealie by Seiden 27. When ~ 5, s > 2, the exact value of ~(,s) is not known. The best uppe bounds cuently known ae due to Tallin1, and Balotti. Thus Tallini 12,7 has shown that -2 ~ (,s ) < s + 1, s > 2, ~ 4 the esult holding both fo odd and even s. When s > 3, Balotti J}, has im.. poved the bound given above. He shows that: (1.6) 2-5. II)(,s) ~ s- - (s-5) E s~ + 1, i=o ~ 5, s ~ 7 and odd,
4 s = 5, -2-6 i (1.8) In; (,s.) ~ s - 26 E S - 1, i=o s = 5, > 6, (1.9) m;(5,s) ~ s3, seven, (1.10) -2-6 i m; (,s ) ~ s - s E s, i=o seven > 6 We obtain in Theoem 2, section 4, a new bound which is an impovement ove these esults when s = 3, o s is even (6 > 2). Fo s = 5, = 5 ou bound is the same as Ba10tti f s. In othe cases Ba10tti f s bound is bette. Fo lowe bounds on ~(,s) and othe impotant esults on non-collinea sets efeence may be made to Sege l. 2. Symmetic epesentation fo t = 3 A set of points in finite pojective space PG(-1,s) of -l dimensions, based on the Galois field GF whee s = ph (p being a pime), is said to be nons collinea set if no thee of the points ae collinea. The set is said to be canplete if we cannot add any new point to the set, so that it still etains the popety of non-co1lineaity. The maximum numbe of points in a non-collinea set in pg(-l,s) may be denoted by m;(,s). The same numbe in the notation used by Balotti [)7 would be denoted by M 1 Let - -,s (2.1) be a complete non-collinea set in PG(-l,s). Since thee ae N = (s - l)/(s-l), distinct points in pg(-l,s), thee ae n = N - m points
5 n = N.. m not contained in the set S given by (2.1). We will denote by' S the set of points (2.;). Fom the popety of non..collineaity no line in pg(-l,s) can intesect the set S in moe than two points. A line 'Will be said to be a secant of S, if it intesects S in two distinct points, it Will be said to be a tangent to S if it intesects S in a single point, and will be said to be a non-intesecto if it contains no point of S. Though each of the points B in (2.;) thee must pass at least one secant i of S. If not ~,A2' Am' B would be a non-collinea set contadicting the i popety of maximality. Let u be the numbe of secants though i B 1 (i = 1,2,...,n = N.. m), 1 ~ u ~ lm/~7, whee 1!7 denotes the Jagest intege i not exceeding!7. On evey secant thee must occu exactly s..l of the points B, since each line of m(-l,s) has exactly 6+1 points. The secants can be exhibited in a tabula fom as follows, whee the i-th ow shows the secants passing though B l ~ll A l12, (2.4) B 2 A 21l A 2l2,, B i Bl ~2l A1221 B 2 ~2l A 222, B A A i i21... i22, B A A, n n2l n22...,...,,...,...,...,..., Hee the points A ijk
belong to the non-collinea set S = ~, A 2,..., Am' and the points B l, B 2,..., B n belong to the complementay set s. The coodinates of B i can be epesented by the vecto 6 and the coodinates of At by the vecto i = 1,2,, n =N - m l' (2.6) t = 1,2,,m The elations between these vectos then can be exhibited as whee A.'S and pis ae non-zeo elements of GF and the a's belong to the. s- set of vectos ~l' ~2'... ~m epesenting the points of S, (i = 1,,2,...,n). The scheme (2p4) o its vecto equivalent (2.7) may be ca.lled the symmetic epesentation fo t = 3. It specifies the stuctue of the complete non-collinea set S = A l, ~,, Am. We shall use this scheme fo obtaining an uppe bound on ~(,s). This scheme can also be used fo constucting non-collinea sets, but this will be consideed in a sepaate pape. 3. Pope a.nd impope non-collinea sets Conside the complete non-collinea set S given by (2.1). A plane cannot contain moe than s+2 points of S when s is even, and moe than s+l points when s is odd, fom the esult due to Bose 2_7 mentioned in the intoduction, since the points of S contained in must themselves fom a non- collinea set. We shall call the non-collinea set S, impope if thee is at least one plane
which contains s+2 points of S. In this case s must be even. We shall 7 call the non-collinea set S pope if thee is no plane of which contains s+2 points of S. In this case s may be odd o even. Let the complete non-collinea set S = AI' A 2,, Am be impope, and let be a plane which contains s+2 points of S, which may without loss of geneality be taken to be Al'~'... AS+ 2 Though the plane thee pass N 3 = (s-3 l)/(s-l), - 3-s:Paces of PG(-l,s). NoW fom the esult of Qvist 117 mentioned in the into- 2 duction, no 3-space can contain moe than s +1 points of S. Hence each 3-space contains s2_s _ l o less points othe than AI' A 2,..., AS+ 2 We theefoe have the theoem: Theoem 1. The numbe of points m in an impope non-collinea set in PG(-l,s), s > 2, satisfies the inequality m~ (s+2) + (s2 - s - 1)N _ 3 4. Some useful lemmas. The numbe u i of the secants of S, passing though B i may be called the weight of (4.1) B J. We may wite Each of the points A, contained in a secant though B. may be supposed to con J. tibute a weight 1/2 to B.. The weight of B. is then the sum of the weights J. J.. of all the points A, lying on secants though B i, i.e. the weight of all the points A occuing in the i-th ow of the scheme (2.4).
Lemma 1. n 1 ~ u i = "2 m(m-l)(s-l), i=l 8 The numbe of distinct secants of the set S is exactly m(m-i)/2 since any secant must contain exactly two points of S. Conside the secant passing though the points Ai and A Thee ae exactly s-l of the points B, on AiA j j The points Ai and A each contibute a weight 1/2 to each of these 6-1 points, j and to no othe points B. Hence the total weight of the points B is m(m-1)(s-i)/2. This poves the Lemma. We s...al1 define the weight of a secant AiA j as the sum of weights of all the 5..]. points B, contained in AiA j, and denote this weight by W(A~j) n 2 Lemma 2. ~ W(AiA j ) = ~ u., i,j = 1,2,, m i<j i=l 1 Conside any point B i Thee ae U i secants passing though it. Hence in counting the 'Wei_ght of' all the secants, the weight of B i is counted u i times. This poves the esult. Lemma. ;. If the non-collinea set S = AI' A,, Am is pope 2 (4.2) tion (4.;) s-l W(A i A j ) ~ ~ W{B ijk ) k=l Lst At be a point of 5 distinct fom Ai and A j If ~ contibutes a. weight 1/2 to B ijk, thee exists a point ~ of S (distinct fom Ai' A j, At) such that the line At~ passes though B ijk If At contibutes a weight 1/2 to each of the points B, '1' B, '2', B. j l' then thee exist distinct points. 1J 1J 1,s- Al' A 2,... A s _ 1 (distinct fom Ai' A j, At) of 5, lying in the plane ~AjAt Hence this plane has s+2 points of S, which contadicts the fact that S is
pope. Henee At can contibute a. weight 1/2 at most to utmost s-2 ot the points B on AiA j Hence the total weight eontibuted by the m-2 points ot S not lying on AtA j to points of! on AiA j does not exceed (m..a)(s-2}/2. On the othe band each ot the points Ai and A j contibutes a weight 1/2 to each of the points Bljl' B 1j2 " H, Blj,S_l' Hence s-l (4.4) 1: w(b ijk ) ~(s-l) + ~ (m-2)(s-2) k=l This. poves the Lemma. 5. Uppe bound on the numbe of R0ints contained in a non-collinea set in o(-l,s), whee 6 > '2 Fist let the non-collinea set s.. ~,~,..., Am be pope and canplete. Then fom LeJmnas 2, and ;, 9 m(m-l) 2 Also ~om Lemma. 1 > n -= N - m 222 mlm-l) (sw1) (N Hence fom (5.1) and (5.2) we have.. m) 2( 2 ( 2 ( m s - s.. 1).. m s - 26.. 1) + N 5-2) -2N < 0 - Hence m cannot exceed the positive oot o the quadatic equation '2 2 x (s.. s - 1).. x 2 (8.. 25.. 1) + N (s..2).. 2 N = 0
i'.. n If the non-collinea set S is impope and s > 2, then 6 = 2, whee 10 n > 2. One can check afte some calculation that in this case the esult of eplacing x if > 4. in the left hand side of (5.4) by (s+2) + (6 2-6 - l)n ~ is negative. -/ Hence the uppe bound on m given by (3.1) is smalle than the uppe bound on m given by (5.5). The late bound is theefoe valid in all cases whee s > 2, ~ 4. If we have a non-collinea set which is incomplete then we can add moe points to it to make it a complete non-collinea set.. Hence we have the theoem: Theoem 2. If Il)(,s) denotes the maximum possible numbe of points in any non-collinea set in o(-l,s), then Il)(,s) cannot exceed the positive oot of the equation (5.4), if s > 2, ~ 4. Hence if s > 2, ~ 4, Il)(,s):5,(,s) whee \!fe,s) i8 given by 2 2 2 3 2 2 1/2 N (8-2)+(S -2s-1)+~N~(S-2) +2N (6-5-2)+(5-2s-1) 7 vc,s) < 2 - - 2(S -s-l) whee N = (s_l)/(s_l).. 6. Discussion of special cases, and compaison with peviously known bounds. (a) Conside the special case s = 3, ~ 50. Let f(x) denote the left hand side of (5.4). Then whee N = {3-1)/2 Let u = (2 + N )/5 Then feu + 1) = 7.. N < 0, feu + 2) = 24 > 0
,,. u Hence,}" (,s ) lies between u + 1 and u + 2. ~(,s) ~,(,s) < N + 12 5.. The bost peviously known bound due to Tallini IE7 :Ls given by 1.5) whieh in this case educes to Hence fo this case the bound given by (5.') is always bette than the bound given by (1.5). (b) Conside the special case s > 3,.. 5. Denoting the left hand side of (5..4) by f'(x) we have 2 '2 t{x) xl:x(s - s.. 1).. (s.. 2s - 1) - N,(s-217.. 2N 5 Let Fo any c '2 (8.. 26.. 1).. N,(e-'2) u :;;: 2 (8 -a-i) 2 feu + e) = (u + c) e (s.. s.. 1).. 2M, '2 2 2 := N fc( a-2).. 2_7 + e ( s.. 28.. 1) + C (s - s - 1) 5 Taking c:= 3(S+1)/(s2 - s _ 1), we see that f(s3) > 0 Taking c = -1 + [)(S+1)/(8 2.. s.. 117, we find that f(s3-1) < 0
,. tt 12 Hence V(IS) lies between s3-1 and 53. We theefoe have This equals the Balotti's bound given by the equation (1.1) fo the case s = 5, and impoves Balotti's bound given by the equation (1.9) when 8 is even. (c) Conside the case when s is even, s > 2 1 > 6..AB befoe let f(x) denote the left hand side of (5.4) and let Then (s2 _ 28-1) - N (s - 2) U = 2 (s - s - 1) feu) = -2N < 0 I f(u+l) = (s-4)n + (s-2)(2s+l) Hence v(ls) lies between U and u + 1. Thus It is easy to check that in this case u+l is less than the ight hand side of the equation (1.10), 80 that ou bound is an impovement ove that given by Balotti.
,.. 13 'l::.7 REFERENCES A. Balotti" "Una limitazione supeioe pe il numeo di punti appalenenti a una calotta (k"o) di uno spazio lineae finito~if Boll. Un. Mat. Ital" 12 (1957)" pp. 67-70. :?:.7 R.. C. Bose, "Mathematical theoy of the symmetical factoial design" It Sankhya, 8 (1947), pp. 107-166. ~7 R. C. Bose, "On some connections between the design of expeiments and info:.mation theoy" If Bulletin de 1 1 Institute Intenational de Statistique, 38" 4 e (1961) pp. 257-271.!!.7 R. A. Fishe, "The theoy of confounding in factoial expeiments in elation to the theoy of goups" It Ann. Eugen. Lend., 11 (1942), pp. 341-353. 27 R. A. Fishe, "A system of confounding fo factos with moe than two altenatives giving completely othogonal cubes and highe powes," Ann. Eugen. Lend., 12 (1945)" pp. 283-290. J R. W. Hamning, "Eo detecting and eo coecting codes, ff Bell System Tech. J., 29 (1950)" pp. 147-160. 17 B. Qvist, "Some emaks concening cuves of the second degee in a finite plane, ti Ann. Acad. Sci. Fenn. see A" I, no. 134 (1952). J B. Sege, "Le geometie di Galois," Ann.. Mat., 48 (1957)" pp. 1-97. 2.7 E. Seiden, IfA theoem in finite pojective geomety and an application to statistics," poc. Ame. Math. Soc., 1 (1950), pp. 282-286. 12.7 G. Ta11ini" "sulla k-ca1otta di uno spazio 1inea.e finito, II Ann. Mat., 42 (1956), pp. 119-164.