Binary sequences with restricted repetitions

Binary sequences with restricted repetitions

Citation for published version (APA):

Post, K. A. (1974). Binary sequences with restricted repetitions. (EUT report. WSK, Dept. of Mathematics and Computing Science; Vol. 74-WSK-02). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1974

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TECHNISCHE HOGESCHOOL EINDHOVEN NEDERLAND

ONDERAFDELING DER WISKUNDE

TECHNOLOGICAL UNIVERSITY EINDHOVEN THE NETHERLANDS

DEPARTMENT OF MATHEMATICS

Binary sequences with restricted repetitions by

K.A. Post

T.H.-Report 74-WSK-02 May 1974

Abstract

Generating functions are given for the number of binary sequences, consisting of r zeros and s ones, no bit occurring more than k times in succession. For k

=

2 a function theoretic analysis is given for the number of sequences containing as many zeros as ones.

- 1

-I. Introduction.

Recently, in feedback communication theory the following coding scheme was considered:

Let k be a fixed integer ~ 2. A message sequence is supposed to be a binary sequence ~n which no k+ 1 successive bits are all of the same parity. This sequence is to be transmitted across a binary symmetric channel with a noiseless, delayless feedback link. The received digits are sent back via the feedback link, so that the transmitter is aware of the transmission errors. Every time a transmission error occurs, a block of k+ 1 repetitions of the correct bit is inserted in the message sequence immediately after the symbol that was wrongly received. Transmission of message sequence plus in-serted correction bits is continued until a given part of the original message sequence is transmitted. The receiver has various decoding

proce-dures at his disposal (cf. [IJ, [2J). Different message sequences turn out to have different sensitivities with respect to channel errors, sequences with (almost) as many zeros as ones being the least sensitive (balanced

sequences). In this paper a recurrence is given for the number of message sequences with prescribed (0,1) inventory. For k = 2 a function theoretic analysis is given for the number of balanced sequences.

2

-II. Mathematical formulation. Elementary results.

Let k be a fixed positive integer. Let S = Sk be the set of all finite binary sequences that contain no k + 1 zeros in succession and no k + 1 ones in succession. More specifically, let A= ~ and B = B

k denote the (comple-mentary) subsets of S consisting of those binary sequences, that start with

a zero and with a one, respectively. Finally, for all r ~ 0, s ~

°

«r,s)

1

(0,0», a and b are defined to be the number of sequences in r,s . r,s

A and B respectively, that contain r zeros and s ones. It is useful to de-fine aOO:=b OO := 1, and ars:=brs:=O if r < 0 or s < 0.

Every sequence in A can be split up uniquely in a starting block of, say j

(1 ~ j ~ k) zeros and a (possibly empty) sequence in B. A similar argument holds for sequences in B, so that

+ ••• + a_r,s-k (1) { ar,s = br-1,s + b r - 2,s b_r,s

=

a_r,s-! + a_r,s-2 + ••• + b r-k,s (r ~ 0, s ~ 0, (r,s)

1

(0,0» .

Define the generating functions a and 8 by 00 00 a(x,y) :=

_{L L}

a xr ys r=O s=O r,s 00 00 8(x,y) :=

_{L L}

b xr ys r=O s=O r,s

Then (I) _{can be restated in the form}

(2) so that

{

"(x.

y) - 1 :

(x

+

x:

8(x,y) - 1 = (y + y + ••• + + ••• + k x )8(x,y) k y )a(x,y) (3) 2 k ( ) l+x+x + ••. +x a x,y

=

----~----~-...".;....--~-2 k 2 k 1 - (x + x + ••• + x ) (y + y + ••• + y ) 2 k Q ( ) 1 + Y + Y + ••• + y ., x,y

=

----~"--₂

....

-:--..,,._~,;,...-~----~_{k 2 k .} 1 - (x + x + ••• + x ) (y + y + ••• + y )

3

-The apparent identity a(x,y)

=

S(y,x) is immediately clear from the fact, that replacing zeros by ones and ones by zeros transforms every sequence of A into a unique sequence of B and conversely, so that a

=

b • This

r,s s,r

argument also enables us to give an explicit construction of the array (a ) in a recurrent way, viz.

r,s r,s (4) a :=

°

r,s aO,O := (r <

°

or s < 0) a r,s as,r-l+ as,r-2+···+ as,r-k (r ~ 0, s ~ 0, (r,s)

:f

(0,0» • A more symmetric recurrence, which also directly follows from (3) may be obtained by applying (4) twice, i.e.

a_r,s := 0 (r < 0 or s < 0) a r,O := (0 ~ r :s; k) a r,O :"= 0 (r > k) (5)

1

a := 0 (s ~ 1) D,s k k

l

ar,s = _{i= 1 j=1}

I

I

ar-i,s-j (r ~ 1, s ~ 1) For k 2, e.g. the array (a ) reads as follows:

r,s r,s 3 0 5 0 4 0 6

°

2 3 4 5 6 0 0 0 0 0 0 0 0 0 0

2 3 2 0 0 _{The array (ar,s)r,s}

2 5 7 6 3 for k = 2 5 12 17 16 10 0 3 13 29 42 42 0 9 33 71 104

o

s 2 r

o

4

-Remark. The number of sequences in A of length n

=

r +s equals F J the n-th n

Fibonacci number (n-th Fibonacci number of order k). This is easily illus-trated by the generating function a(t, t) = (1 - t - t2- .•• - tk) - I.

An interesting subset of A is formed by the balanced sequencesJ i.e.

se-quences for which r = s. Their number corresponds with the number of paths in an s x s square from the left bottom vertex to the right top vertex, that have minimal length, consist of only horizontal and vertical segments of integer length ~ k and start in horizontal direction. For arbitrary k the analysis of these numbers is hard. For k

=

2, howeverJ a generating function

and a recurrence relation can be found explicit~y. For k

=

2 the numbers are

• 00

found on the d~agonal (d) 0 of the array (a ) and read as follows:

S s= r,s r,s

00

5

-III. Function theoretic analysis for the case k = 2. The double series

00 00

L L

r=O s=o r s a x y r,s (cf. (5))

is absolutely and uniformly convergent for complex x and y, Ixl = 1, Iyl ::;

!

(/3 -

1), hence the integral

1 2ni

J

w-1 Iwl=1 z a(w ,-)dw w 1

=

_2ni 00

L

s=O a_r,s wr-s-l zs dw

may be calculated by term-by-term integration and arbitrary order of summa-tion and has the value Loo a ZS for complex z of sufficiently small

s=O ss absolute value.

On the other hand (cf. (3)) 1 2ni

J

w-1 Iwl=1 z a(w ,-)dw = w w -I 2ni

_J

Iwl=1 I + w + w2

---_=_

dw (l +w) (zw+z2) dw ,

where wI and w₂ are the roots of the quadratic equation

so

2 3 4 1

+ (I - 2z - z - 2z + z )2 ]

1 2 2 3 4!

(Zz) - [- Z - Z + I - (1 - 2z - z - 2z + z )2 ]

For small z the root wI is outside and w

2 ~s inside the unit circle, so that by the residue theorem out integral has the value

2

w

2 + W

z

+ I z(w

6

-which can be written in the form

(6) to to

L

d_s zs =

_L

a zs = s=O s=O ss 1 _{+ -}1 _{+.!. ( -}1

-2z2 2 2 z2 z 1 1 1 2 2 1 2 _I = - --- + - + - - (1 - z) (1 + z + z )2(1 - 3z + Z ) 2 2z2 2 2z2 where 2 . - -1T1 3 e

Corollary. Since this function has z4 as branch point of smallest absolute value, it follows that d asymptotically behaves as the Taylor coefficients

1 s _1

of (1 - :4)-2. By Stirling's formula this yields d

s ~ D s 2 F2s, D being a constant, F_2s a Fibonacci number.

It is also possible to obtain a recurrence relation for d from (6). For

s

this purpose we write

1 1 1 2 2 1 2 _1 G(z) : = - --- + - + - - - (I - z) (l + z +Z ) 2(l - 3z+ Z ) 2 , 2z2 2 2z2 so that 2 2 2 2 1 2 _1 2z G(z) + 1 - z = (1 - z) (1 + z+ Z ) 2(1 - 3z + Z ) 2 ,

and, by logarithmic differentiation 2 4z G+ 2z G'

-

2z

-

2 1 + 2z 1 3 - 2z 2 2 1

-

z + -2 2 +'2 2

,

2z G + 1 - z + Z + Z - 3z + z 2G + zG'

-

1 + 3z2 - z3 2z2 _G + I

-

z2 I - 3z + z2

-

z3 _{+ 3z}4 - z5

,

7

-Substitution of G(z)

= r:=O

d

s zS yields, by identification of coefficients

(7)

J

:::2:d' _d:<::':d d21:

:n~2:: =27_'<n~4J~d173'+

l

n n- n-

n-+ 3(n - 4)d

n_4 - (n - 5)dn_5 = 0 (n ~ 5) •

References.

[IJ J.P.M. Schalkwijk, A class of simple and optimal strategies for block coding on the binary symmetric channel with noiseless feedback.

IEEE Trans. Inform. Theory, vol. IT-17, pp. 283-287, May 1971.

[2J J.P.M. Schalkwijk and K.A. Post, On the error probability for a class of binary recursive feedback strategies. IEEE Trans. Inform. Theory, vol. IT-19, pp. 498-511, July 1973.

Note added in proof:

In a recent paper (The Fibonacci Quarterly, Vol. 12, No I, 1974, p. 1-10) L. Carlitz gives generating functions like (3) for the slightly more general situation where no k + I successive ones and no R, + I successive zeros are allowed.