ON BINARY SEQUENCES1 R. TURYN AND J. STORER

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1 ON BINARY SEQUENCES R. TURYN AND J. STORER Let X ±, l =,, ra and let ck = 23?"* *»*»+/. We shall be concerned wth sequences (x) f whch \ck\ ál 0< <w. The problem was frst consdered n [l]. The followng sequences wth ths property are known: ra= ; where t has been assumed x = x2= +. Each one of these gves rse to three others under the transfmatons () x x = (-!)<*, = (-l)í+*, The above sequences have the followng nterpretaton [2]: the elements whch are are those wth subscrpt gven mod ra by ra = 5: m + 3, m ^á 0 (mod 5), ra = 7: m2 + 3, m fé 0 (mod 7), n = : m 2, (mod ), ra = 3: ID»4 -, (mod 3). It wll be shown here that f odd ra, these are all the sequences wth the property \ck\ =. F the sake of completeness, the prelmnary detals have been ncluded. Let (x) be any sequence wth *<== ±, and let Receved by the edts May 26, 960. The research repted n ths paper has been partally sponsed by the F.lectroncs Research Drectate of the AFCRC, Ar Research and Development Command, under Contract AF-9(604) Lcense copyrght restrctons may apply to redstrbuton; see

2 on bnary sequences 395 n k Ck = X XX+k. Snce the terms x&+u are each ± I, t follows that : (2) IT (*«*»*) - (-!)<»-*-«*>/*, 0 ^ <». Multplyng two successve equatons of ths fm gves Now consder Xn-kXk+l = (-l)"-*-(l+<*+c*+l)/2; á * á» -. n (the second ndex beng taken mod re). Then (2) yelds n II (XX+k) = = (-) <»-<*-< -*)/2_ Thus Ck-\-Cn-k n (mod 4). We now assume [c* l, 0<k<n; thus c* = 0 f n k even, Cfc= ± f n k odd.2 Now assume also that re s odd. In ths case, one of Ck and cn-k s odd and the other case even, hence 0. Thus c2+ = 0, Cy=(-l)<-m the latter snce Ck-\-cn-k n (mod 4). The fmula f xn-kxk+ becomes (3) xn-kxk+ = (_l)<»->/!+*. A sequence whch satsfes (3) has the property that c2y+ = 0, 0< 2j + l<«. n 2j coy = 23 *<#<+«! 0 < 2/ < re, (_l)(n-l)/2 Let n-2j = 2k + l, k^l. = ".JSUS^^^-D^O/Wfl. 2 It follows from ths that» = 2(mod 4) mples n = 2. F f n>2, C2 = c _2 = 0, C2+c, _2 = 0^2 (mod 4). Thus n even, n>2 mples»^0(mod 4), *+!:»-* "». Lcense copyrght restrctons may apply to redstrbuton; see

3 396 R. TURYN AND J. STORER June = 2*+l Y, XX2k+2-.(-y+ Hence = 2 ^2+2_,-(- l)+ + xl+(-)*+2, 02HU3. l + t-l)^ * ra- (k) \ = E ***+!-,(-U" = * < -r 2 2 We shall wrte P(k) f (l + (-l)*+)/2. Theem. Let (x) satsfy equatons (k) f ^k^t, X= +. Le X=\ f l^úp, xp+= ; assume p>\. Then () *íx,+ = ;*;2,-x2í+, lááí, () p'è.2t + \ mples p s odd, () pj+r^2t +, l^r^ mples xvy-)+t = xp<j-)+, and (v) 2, = x;py_)+ satsfy equatons (k) f k^t/p. Proof. A count of +l's and l's n equaton (k) shows as befe that t l (XXk+^) = 2t+l n X = **+!. Multplyng two successve equatons of ths fm proves the frst statement. The second statement follows from the frst: f f p>2 were even, say p = 2s, we would have X,X,+ = XfrX^+, by assumpton, =. We prove () and (v) by nducton on /. From t<p, there s nothng to prove. F p = t, we have : (p) = tfl*2jh-l X2X2p + + XpXp+2. Now X=\ f up, and by the frst part of the theem we have x2k = x2k+ f p<2k<2p. Thus the above equaton reduces to But also = x2p+ x2p + xp+2. Lcense copyrght restrctons may apply to redstrbuton; see

4 96] ON bnary sequences 397 so we have Thus = XpXp+ = X2pX2p+, = 2x2p+ + Xp+. x*+* = -, X2p+ =. Smlarly, f 2+l ^ 3p, we let t = p+s, s>0; we have 2«+l P P+ë P(P + S) = J2 + J2 + Z) = Xlp+l - Xp+Xp+u+l + P(s - ), 2+2 jm- as- usng the fact that x2 = x2+ f ^O (mod p) by the nducton sumpton and (). Snce p s odd, P(p-\-s)=P(s l), and *2jH-l + Xp+u+l = 0 Xp+,+l =. F 2+l>3, we let 2t+l = hp+m wth l<m^p, and consder two cases: (A) A odd, A = We have t H l r p+m (}'+l)p "I HP+m t Z=E + S + S + E. 0 L yjh-l j'jh-m+l J ffp+ J/p+m+ As befe, we reduce the frst three sums by usng the fact that x2 = X+ f jé 0 (mod p) ; the subsequent sums are of the fm ( ) ' by the nducton assumpton. Thus we get P(t) = XfX^-Dp+m+l Xp+Xyt-Dp+m + jh-mz(ä-l)jh-l We have t = Hp + (p+m-l)/2. + e ^*+-.(-)<+ + (-)hp(- ^ y Therefe H P(B) E ZZk+l-( l)+ = X^-Dp+m+l + X(H-)p+m 2X(k-l)p+l. By the nducton assumpton, the left-hand sde vanshes, and thus #(A-l)p+m = (»-l)p+l. (B) If A s even, A = 227, we get, proceedng as befe, Lcense copyrght restrctons may apply to redstrbuton; see

5 398 R. TURYN AND J. STORER [June H P(H) X] ZZh+-( Y)+l = zh xhp + 2xh-)p+m f m < p. Subtractng successve equatons of ths type (.e., m=, 3, ) gves xkh-)p+=x^-v p+m (f m odd, and the equaton f m even follows from ()). F m = p,.e., 2t + \ (2H+\)p we get H P(H) - z2,l+_,-(-l)+ = Zh + xhp. Comparng ths wth the equaton f m =, we deduce and hence B P(H) - Zz^+-(-l)+ = 0 XhP ~ X^h-Djt+l. We have shown that a sequence of odd length wth \ck\ è f k>0 has some perodc behav. We now conclude that t must therefe be sht. Theem 2. Let x = ±, l S un, n odd. Assume \ck\ = f k>0. Then ra ^ 3. Proof. By usng the transfmatons (), we may assume that x = x2=l. We have equatons (k) f l^ <(ra )/2, and clearly p (as defned n Theem ) s less than ra f ra>3. Frst, we must have n<4p: f f n>4p, we would conclude by Theem that X=+l, lúúp, 2p + í^^3p and x,= -l, p + \^ú2p. But f X = X+, x +-= Xn- by (3), and runs of equal X are opposte runs of alternatng ones. n>4p would mply 3 ntal constant blocks, hence 3 fnal alternatng blocks, and therefe n^6p. Smlarly, f ^2 Theem shows that n>kp mples ra :2( V)p 3. Thus n<4-p. We also know n^3p, f n = 3p would mply XpXp+ = X2pX2p+ Xp+Xp. Let o= ( l)(n_i)/2, ra 3 = 2?ra. Then equaton (m) s \ + a - = 2- *Ä_-.-(-l)+. The frst p 2 terms n the sum are equal to o. Let N be the number of terms x,-x ( l)+ = a wth p 2<^m. Then Lcense copyrght restrctons may apply to redstrbuton; see

6 g6] ON bnary sequences 399 a - = (p - 2)a + (m - N - (p - 2))a - Na, - + N - (p - 2) = m - N - (p - 2) = 0, a + «3 p-2- g N = _ - ( - 2), 4/> =» a. Hence n<3p mples p<6+a p^5. By Theem, we can construct the sequences of lengths 5, 7, 3. Fnally, f 3p<n<4p, let re = 2w + l. xm= xm+2, and we have x<= f ^ + l=já2^. By Theem, we must have n 2 p m-\-l = (re + l)/2, w^2 +5. Now n>3p mples p<5, and hence p = 3 and we reconstruct the sequence of length. References. R. H. Barker, Group synchronzng of bnary dgtal systems, n Communcaton they, London, 953, pp Marshall Hall, Jr., A survey of dfference sels, Proc. Amer. Math. Soc. vol. 7 (956) pp Sylvana Electrc Products, Inc. Lcense copyrght restrctons may apply to redstrbuton; see