The sizes of consecutive repeat-free codes

Robin Hughes-Jones

(2010)

Robin Hughes-Jones (2010) The sizes of consecutive repeat-free codes.

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Abstract

The notions of strongly consecutive repeat free code and weakly consecutive repeat free code were introduced by Pebody in his paper in the Journal of Combinatorial Theory Series A in 2006. This thesis aims to investigate the the maximum sizes of such codes, in particular in the case when the length is fixed and the alphabet size is large. Pebody constructs a strongly consecutive repeat free code of maximal size, which he calls the alternating code. We show that the size of an alternating code is polynomial in the alphabet size, give methods for computing this polynomial and explicitly determine the most significant coefficients of this polynomial in terms of the sequence of `up/down numbers' and related sequences. Pebody defines a family of codes (which we call Pebody codes) that are weakly consecutive repeat free codes. Pebody conjectures that for all parameters there exists a member of this family that is a weakly consecutive repeat free code of maximal size. We show that the maximal size of a Pebody code agrees closely with the maximal size of a strongly consecutive repeat free code. We use techniques from combinatorics and functional analysis, together with computational results, to give estimates for the leading terms of the maximal size of a Pebody code of fixed length when the alphabet size is large.

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This is a Published version
This version's date is: 02/03/2010
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https://repository.royalholloway.ac.uk/items/acd01faa-158b-5e71-43b1-93fd1b5221bc/1/

Item TypeThesis (Doctoral)
TitleThe sizes of consecutive repeat-free codes
AuthorsHughes-Jones, Robin
DepartmentsFaculty of Science\Mathematics

Deposited by () on 23-Jun-2010 in Royal Holloway Research Online.Last modified on 15-Feb-2017

Notes

References

Alan F Beardon. Sums of powers of integers. The American Mathematical
Monthly, 1996.

Bruce Bauslaugh and Frank Ruskey. Generating alternating permutations
lexicographically. BIT, 30(1):17{26, March 1990.

Yuliy Baryshnikov and Dan Romik. Enumeration formulas for Young
tableaux in a diagonal strip. Israel Journal of Mathematics, 2009. forthcoming.

Lennart Carleson. On convergence and growth of partial sums of Fourier
series. Acta Mathematica, 116(1):135{157, December 1966.

William Dunham. Euler: the master of us all. Mathematical Association
of America, 1999.

Paul Richard Halmos. Introduction to Hilbert space and the theory of spec-
tral multiplicity. Chelsea publishing company, second edition, 1957.

 Donald E Knuth and Thomas K Buckholtz. Computation of tangent, Euler
and Bernoulli numbers. Mathematics of Computation, 21(100):663{688,
October 1967.

Serge Lang. Complex analysis. Springer, fourth edition, 1999.

Ronald Larsen. Functional analysis an introduction. Marcel Dekker, 1973.

Blaise Pascal. Sommation des puissances numeriques. 1654.

Luke Pebody. The largest strongly consecutive repeat-free code. Journal
of Combinatorial Theory, Series A, 113(3):551{555, Apr 2006.

N J A Sloane. An on-line version of the encyclopedia of integer sequences.
Website. http://www.research.att.com/~njas/sequences/.

Edward Charles Titchmarsh. The theory of functions. Oxford University
Press, second edition, 1939.

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