On Counting Sequences and Some Research Questions

I Nengah SupartaMathematics Education DepartmentMath. and Natural Sciences Faculty

Ganesha University of EducationSingaraja – Bali, Indonesia

Outline

• Motivation (applications)• Definition of counting sequences• Types of counting sequences• Some facts and related unsolved problems

1. MotivationApplications of Uniform counting sequences (Gray codes):

• analog to digital information conversion• error correction• circuit testing• signal encoding• data compression• diagnosis of multiprocessors• computational biology

[Hayes; Knuth; Robinson and Cohn]

2. Basic Definitions and NotationsDef.2.1 A string x = xnx(n – 1) … x1

where xj ϵ {0, 1} for all j, 1 ≤ j ≤ n, is called a binary codeword of length n or simply n-codeword. A counting sequence-n is an ordered list of all 2n distinct n-codewords. Index of codewords: from 0 until 2n – 1;Bit positions: counted from right to left from 1 to n. Codeword of index i, written as xi = xinxi(n-1) … xi1.

If x0 and xp, p = 2n, share the property as imposed to any two consecutive codewords, then the sequence is called cyclic. In this case, the codewords x0 and xp are identified.

• Hamming distance of two n-codewords x = xnxn-1...x2x1 and y = ynyn-1...y2y1, is

dH (x, y) = #{i| xi≠yi, i =1, 2,..., n}

• List distance of xi and xj in a (cyclic) counting sequence-n is

D(xi, xj ) = min{|i – j|, 2n – |i – j|}.

Example of (cyclic) counting sequences0000000100110010011011101010101110011101111101110101010011001000

a

0000110101101011110000100101100000110100111100011010011110011110

b

0000111100011110001111000010110101101001011110000101101001001011

c

Example of (cyclic) counting sequences0000000100110010011011101010101110011101111101110101010011001000

a

0000110101101011110000100101100000110100111100011010011110011110

b

0000111100011110001111000010110101101001011110000101101001001011

c

6

Def.2.2 Let C := x0, x1, … xp–1 be a counting sequence-n with p = 2n, and let

si := {j|xi–1 j ≠ xij}, and

TC(j) := #{i|jϵ si, 1≤ i ≤ n}.

In a counting sequence-n, the sequence s1, s2, …, sp, is called the transition sequence of the sequence-n, and the distribution

TC = (TC(1), TC(2),…, TC(n))is called transition count spectrum of the sequence-n. (TC(1)/2n, TC(2) /2n,…, TC(n) /2n) is referred to as the bit error probability of the sequence.If |TC(i) – TC(j)|≤ 2, for every i,j, then the sequence is called balanced, and it is called totally balanced if |TC(i) – TC(j)|= 0 for every i,j.

Example of cyclic counting sequences0000000100110010011011101010101110011101111101110101010011001000

a

0000110101101011110000100101100000110100111100011010011110011110

b

0000111100011110001111000010110101101001011110000101101001001011

c

<- {1}= s1

<- {2}= s2

<- {1}<- {3}<- {4}<- {3}<- {1}<- {2}<- {3}<- {2}<- {4}<- {2}<- {1}<- {4}<- {3}<- {4}= s15

b

<- {1,3,4}= s1

<- {1,2,4}= s2

<- {1,3,4}<- {1,2,3}<- {2,3,4}<- {1,2,3}<- {1,3,4}<- {1,2,4}<- {1,2,3}<- {1,2,4}<- {2,3,4}<- {1,2,4}<- {1,3,4}<- {2,3,4}<- {1,2,3}<- {2,3,4}= s15

b

<- {1,2,3,4}<- {2,3,4}<- {1,2,3,4}<- {1,3,4}<- {1,2,3,4}<- {2,3,4}<- {1,2,3,4}<- {1,2,4}<- {1,2,3,4}<- {2,3,4}<- {1,2,3,4}<- {1,3,4}<- {1,2,3,4}<- {2,3,4}<- {1,2,3,4}<- {1,2,4}

b

3. Some facts and related problems

Def. 3.1 Let S be the transition sequence of a counting sequence G. If for all si ϵ S, #{si} = 1, then G is called a Gray code.

Def. 3.1.1 Let G be a Gray code and S = s1, s2,… , be the transition sequence of G, then the minimum run length of G is

mrl(G) = min{|i – j|: si = sj, i ≠ j}.

Let mrl(n) be the maximum mrl(G) for every code G of length n.

3.1 Gray codes with maximum run length

The values of mrl(n) for every n < 8 are known to be 1, 2, 2, 2, 4, 4, 5 for n equals 1, 2, …, 7 respectively.

3.1 Gray codes with maximum run length

• Theorem 3.1.2(Goddyn and Gvozdjak) For integers n ≥ 2 we have mrl(n) ≥ .

• Problem 3.1.3. Does a function B exist such that for every n ≥ 2,

mrl(n) ≥ B(n)> ?

Goddyn and Gvozdjak(2003) produced 10-bit Gray code of mrl(10) = 8.

3.2. Gray codes and their Separability capacity

• Theorem 3.2.1(Cavior, Yuen) The separability function of the binary standard Gray code is

b(m) = . • Theorem 3.2.1(van Zanten-Suparta) The

separability function of the N-ary standard Gray code is

b(m) = .

3.2. Gray codes and their Separability capacity

• Park and Bose (2004) constructed a class of Gray codes of length n having separability function

b(m) =

For m even, this is better than the stadard Gray codes, but worse otherwise.

• Suparta (2006) constructed a class of Gray codes of length n with separability function

b(m) =

3.2. Gray codes and their Separability capacity

• Problem 3.2.3 Does there exist a Gray code and a bound b(m), such that if dH(xi, xj) = m then D(xi, xj)

> b(m) ≥ for all m-values with 2 < m ≤ n?

A weaker version of the above formulation is the following.

• Problem 3.2.4 Does there exist a Gray code and a

bound b(m), such that at least for two m-values ifdH(xi, xj) = m, then D(xi, xj) ≥ b(m) > ,

and D(xi, xj) ≥ otherwise, with 2 < m ≤ n?

3.3. Gray codes with prescribed count spectrum

• Theorem 3.3.2(Evdokimov, Knuth) Let (TC(1), TC(2), …, TC(n)) be the transition count spectrum of a cyclic Gray code-n which is ordered in non-decreasing way, i.e. TC(i) ≤ TC(i+1), for all i, 1 ≤ i ≤ n – 1. Then, one has

, for all k, 1 ≤ k ≤ n.

Def.3.3.1 Let the sequence s1, s2, …, sp, be the transition sequence of the sequence-n, p = 2n, and consider the count spectrum

TC = (TC(1), TC(2),…, TC(n))with TC(j) := #{i|jϵ si, 1≤ i ≤ n}.

3.3. Gray codes with prescribed count spectrum

Def. 3.3.3 Let p be a positive integer and p1 + p2 + … + pk = p, where pi is a positive integer for all i {1, 2, …, k}. The distribution or spectrum A = (p1, p2, …, pk) is called a partition of p. Furthermore, if pi is even for all i, the spectrum A is called an even partition of p.

Problem.3.3.4 Let A = (p1, p2, …, pn) be an even partition of 2n

, pi ≤ pi+1, for all i, 1 ≤ i ≤ n – 1,

and satisfy the condition , for all k, 1 ≤ k

≤ n. Does there exist a cyclic Gray code of length n having transition count spectrum A?

3.3. Gray codes with prescribed count spectrum

Theorem 3.3.5(Suparta) Let G be a Gray code of length (n – 2) with transition count spectrum (TC(1), TC(2), ..., TC(n – 2)) and A = (p1, p2, …, pn) be an even partition of 2n. A Gray code of length n with transition count spectrum A exists if and only if (i) pk = pk+1 for some k ϵ {1, 2, ..., n – 1};

(ii) 2TC(i) ≤ pi ≤ 4TC(i) for every i ϵ {1, 2, ..., k – 1},

2TC(i) ≤ pi+2 ≤ 4TC(i) for every i, k ≤ i ≤ n – 2,

(iii) there is at least one i0 ϵ {1, 2, ..., n – 2} such that either 4(TC (i0) – 1), i0 ϵ {1, 2, ...,

k – 1} or ≤ 4(TC(i0) – 1), k ≤ i0 ≤ n – 2.

3.4 Graphs induced by Gray codes

Def. 3.4.1 Let S(n) := s1, s2, …, be the transition sequence of a Gray code G. The graphs ГG induced by the Gray code G has vertex set {1, 2, ..., n} and edge set {{si, si+1}: i ϵ {1, 2, ..., 2n – 1}}.

• It is easy to observe that the standard Gray code of length n induces a star or the complete bipartite graph K1,n.

• Suparta (2006) proved that there exists Gray code of length n inducing complete graph of n vertices. This result constitutes an answer of Wilmer and Ernst conjecture (2002).

3.4 Graphs induced by Gray codes

• Suparta (preprint) constructed Gray codes inducing the complete bipartite graph Km,n for any pair of positive integers m and n.

• An instance, the Gray code which has transition

sequence 2, 1, 2, 3, 4, 3, 2, 3, 4, 1, 4, 3, 2, 3, 4, 3 induces the complete bipartite graphs K2,2. This extends the results of Wilmer and Ernst (2002).

3.4 Graphs induced by Gray codes

• Problem 3.4.2 Does there exist an n-bit Gray code, for every n ≥ 6, which induces the complete graph Kn, and which has no bi-directional edges?

• Problem 3.4.3 Does there exist an n-bit Gray code, for every m, n ≥ 5, which induces the complete bipartite graph Km,n, and which has no bi-directional edges?

4. On balanced sequence-(n, t)

• Gray codes are examples of counting sequences with t = 1. Balanced Gray code of ength n exist for all n. Thus, balanced sequence-(n, 1) exists for any value of n.

Conjecture 4.1 (Robinson-Cohn (1981)) For every realizable pair of n and t, 1 ≤ t < n, a balanced sequence-(n, t) exists.

4. On balanced sequence-(n, t)

Theorem 4.2(Suparta (2006) For every m ≥ 1, there exists a balanced uniform sequence-(2m, 2m – 1), and if m is a power of two, there exists a totally balanced uniform sequence-(2m, 2m – 1).

Theorem 4.3 (Suparta (2006)) For any realizable pair of n and t, with gcd(n, t) = 1, there exists a balanced sequence-(n, t).

Problem 4.4 Does a balanced sequence-(n, t) exist if gcd(n, t) ≥ 3?

References[1] Cavior, S.R., 1975. “An upper bound associated with errors

in Gray code,” IEEE Trans. Inform. Theory, vol. IT-21, p. 596.[2] Evdokimov, A.A., Novosibirsk University, Russia, private

communication.[3] Goddyn, L. and Gvozdjak, P. 2003. Binary Gray codes with

long bit runs. The Elect. J. Conbinatorics. Vol. 10, paper #R27.

[4] Knuth, D.E., 2005. The Art of Computer Programming, Volume 4, Addison-Wesley as part of “fascicle” 2, USA.

[5] Park, J.P. and B. Bose, 2003. “Separabilities of binary Gray codes designed over Z4,” IEEE International Symposium on Inform. Theory, June 29, - July 4.

[6] Robinson, J.P. and Cohn, M. 1981. Counting sequences. IEEE Trans. Computers. Vol. C-30. No. 1. pp. 17-23.

[7] Suparta, I N. 2006. Counting sequences, Gray codes, and Lexicodes. PhD dissertation. Delft Univerity of Technology, the netherlands.

[8] Suparta, I N. and van Zanten, A.J., 2008. A construction of Gray codes inducing complete graphs. Discrete Mathematics. Vol. 308 pp. 4124-4132.

[9] Suparta, I N. 2014. Gray codes inducing complete bipartite graphs. Preprint.

[10] Wilmer, E.L. and Ernest, M.D. 2002. Graphs induced by Gray codes. Discrete Math. Vol 257. pp. 585-598.

[11] Yuen, C.K., 1974. “The separability of Gray code,” IEEE Trans. Inform. Theory, vol. IT-20, p. 668.

[12] van Zanten, A.J., and I N. Suparta, 2003. “The separability of standard cyclic N-ary Gray codes,” IEEE Trans. on Inform. Theory, vol. 49, pp. 485-487.