1999 IEEE ITW. Kruger National Park, South Africa. June 20
- 25
New Results on multiple insertion/deletion correcting codes W.A. Clarke BSW Telecoms Private Bag X35 Halfway House, 1685 South Africa Email
[email protected]
Prof H.C. Ferreira Cybernetics Lab Rand Afrikaans University Auckland Park, 2006 South Africa Email
[email protected]
Abstract - In this paper, new results on multiple symbol insertion and/or deletion correcting codes are presented. Firstly, new relationships between codebook properties and the symbol insertion and/or deletion correcting ability of a codebook are established. Secondly, new results on the upperbound of symbol insertion and/or deletion correcting codebook cardinalities are presented.
I. INTRODUCTION The Levenshtein distance is the number of symbol insertions and/or deletions necessary t o transform codeword x into y [l]. Let s indicate the symbol insertion and/or deletion correcting ability of a codebook Q. In this paper, only block codes are considered with n symbols per codeword x. The Levenshtein distance L of a codebook Q is related to s by:
L
> 2s.
(1)
This study assumes no additive errors are present.
11. INSERTION/DELETION ABILITY
OF A CODEBOOK
111. UPPERBOUNDS OF
Proposition 1 An upperbound on the insertion/deletion correcting ability of any subset Q; is an upperbound on the insertion/deletion correcting ability of the codebook Q. By carefully selecting these subsets to conform to certain code properties, one is able to determine the maximum insertionldeletion correcting ability of a codebook. This is useful to investigate known codes for their insertion/deletion correcting ability. The following result, based on the number of runs , be used with the in codeword x and indicated by ~ ( z ) can result of Proposition 1.
Proposition 2 Let Q be an s insertion/deletion correcting binary code) i for all x E &. Then the following is true book with ~ ( x = for Q: ssn-i. (2) The following Proposition relates s to cyclic shifts of codewords. Proposataon 3 In any s-insertion/deletion correcting codehook &, the minimum number of cyclic shifts between codcwords z and y, z,y E Q must be s + 1.
I2X
INSERTION/DELETION
CORRECTING CODEBOOKS This section lists new results on the upperbounds of insertion/deletion correcting codes.
Proposition 4 Let Q be a q-nary s-insertion/deletion correcting codebook with codewords of length n. Let C(q,n,s)be the cardinality of Q and A(q,n,dmin)the upperbound on the number of codewords that has a minimum Hamming distance of dm;n. Then the following relationship holds for the codebook Q:
C ( q ,n, s) 5 A ( q , n, s
+ 1).
(3)
Values for A(q,n,dmin)are tabled in [2].
Proposition 5 The following recursive equation is true for upperbounds of insertion/deletion correcting codes:
B ( q ,n
Any codebook Q with a cardinality of C, where C 2 2, can be partitioned into several smaller subsets Q, C Q with cardinalities C; < C. The following result enables one t o use these subsets t o derive bounds on the s ability of the code Q.
0-7803-5268-8/99/$10.00@ I999 IEEE
T.G. Swart Cybernetics Lab Rand Afrikaans University Auckland Park, 2006 South Africa Email
[email protected]
+ i , s + i) IB ( q ,n, s)
(4)
where i = 1,2,..., B(q,n,s) is the upperbound of an sinsertion/deletion correcting codebook Q over an alphabet with q symbols and codewords of length n. We further present tables listing cardinalities for multiple insertion/deletion correcting codes utilising the above theories.
REFERENCES [I] V. I. Levenshtein, “Binary codes capable of correcting deletions, insertions and substitutions of symbols,” Doklady Academy of Sciences of the USSR, vol. 163, No. 4,pp. 845-848, 1965. [2] F.J. MacWilliams and N.J.A. Sloane, The theory of errorcorrecting codes, North Holland, New York, 1977
