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Properties of Codes in the Johnson Scheme

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Properties of Codes in the Johnson Scheme Properties of Codes in the Johnson Scheme Natalia Silberstein Properties of Codes in the Johnson Scheme Research Thesis In Partial Fulfillment of the Requirements for the Degree of Master of Science in Applied Mathematics Natalia Silberstein Submitted to the Senate of the Technion - Israel Institute of Technology Shvat 5767 Haifa February 2007 The Research Thesis Was Done Under the Supervision of Professor Tuvi Etzion in the Faculty of Mathematics I wish to express my sincere gratitude to my supervisor, Prof. Tuvi Etzion for his guidance and kind support. I would also like to thank my family and my friends for thear support. The Generous Financial Help Of The Technion and Israeli Science Foundation Is Gratefully Acknowledged. Contents Abstract 1 List of symbols and abbreviations 3 1 Introduction 4 1.1 Definitions.................................. 5 1.1.1 Blockdesigns............................ 5 1.2 PerfectcodesintheHammingmetric. .. 7 1.3 Perfect codes in the Johnson metric (survey of known results)....... 8 1.4 Organizationofthiswork. 13 2 Perfect codes in J(n, w) 15 2.1 t-designs and codes in J(n, w) ....................... 15 2.1.1 Divisibility conditions for 1-perfect codes in J(n, w) ....... 17 2.1.2 Improvement of Roos’ bound for 1-perfect codes . 20 2.1.3 Number theory’s constraints for size of Φ1(n, w) ......... 24 2.2 Moments .................................. 25 2.2.1 Introduction............................. 25 2.2.1.1 Configurationdistribution . 25 2.2.1.2 Moments. ........................ 26 2.2.2 Binomial moments for 1-perfect codes in J(n, w) ........ 27 2.2.2.1 Applicationsof Binomial momentsfor 1-perfect codes in J(n, w) ........................ 31 2.2.3 Binomial moments for 2-perfect code in J(2w,w) ........ 34 2.2.3.1 Necessary conditions for the existence of a 2-perfect code in J(2w,w) .................... 38 2.2.4 Binomial moments for e perfect code in J(2w,w). ....... 44 − 3 Perfect doubly constant weight codes 48 3.1 Definitions and properties of perfect doubly constant weightcodes . 48 3.2 Family of parameters for codes whose size of sphere, Φ1(n1,w1, n2,w2), dividesthesizeofwholespace . 50 3.3 Necessary condition for existence of an e-perfect doubly constant weight code..................................... 51 4 Steiner Systems and doubly Steiner Systems 55 4.1 Definitionsandknownresults. 55 4.2 Necessary conditionforexistenceof Steinersystem . ........ 56 4.3 DoublySteinersystem .. .. .. .. .. .. .. .. 58 Appendix A 61 Appendix B 64 Appendix C 72 Bibliography 76 List of Figures 1.1 2-(7,3,1)design ............................... 6 List of Tables 2.1 w w + a 1(mod12) .......................... 33 ≡ ≡ 2.2 w w + a 7(mod12) .......................... 33 2.3 nonexistence≡ ≡ of 2-perfect codes in J(2w,w) for n< 2.5 1015 ...... 43 ∗ Abstract Codes which attain the sphere packing bound are called perfect codes. Perfect codes al- ways draw the attention of coding theoreticians and mathematicians. The most important metrics in coding theory on which perfect codes are defined are the Hamming metric and the Johnson metric. While for the Hamming metric all perfect codes over finite fields are known, in the Johnson metric it was conjectured by Delsarte in 1970’s that there are no nontrivial perfect codes. The general nonexistence proof still remains the open problem. Constant weight codes play an important role in various areas of coding theory. They serve as building blocks for general codes in the Hamming metric. One of the applica- tions of constant weight codes is for obtaining bounds on the sizes of unrestricted codes. In the same way as constant weight codes play a role in obtaining bounds on the sizes of unrestricted codes, doubly constant weight codes play an important role in obtaining bounds on the sizes of constant weight codes . In this work we examine constant weight codes as well as doubly constant weight codes, and reduce the range of parameters in which perfect codes may exist in both cases. We start with the constant weight codes. We introduce an improvement of Roos’ bound for 1-perfect codes, and present some new divisibility conditions, which are based on the connection between perfect codes in Johnson graph J(n, w) and block designs. Next, we consider binomial moments for perfect codes. We show which parameters can be excluded for 1-perfect codes. We examine 2-perfect codes in J(2w,w) and present necessary conditions for existence of such codes. We prove that there are no 2-perfect codes in J(2w,w) with length less then 2.5 1015. Next we examine perfect doubly constant∗ weight codes. We present properties of such codes, that are similar to the properties of perfect codes in Johnson graph. We present a family of parameters for codes whose size of sphere divides the size of whole space. We then prove a bound on length of such codes, similarly to Roos’ bound for perfect codes in Johnson graph. Finally we describe Steiner systems and doubly Steiner systems, which are strongly 1 connected with the constant weight and doubly constant weight codes respectively. We provide an anticode-based proof of a bound on length of Steiner system, prove that dou- bly Steiner system is a diameter perfect code and present a bound on length of doubly Steiner system. 2 List of symbols and abbreviations n k binomial coefficient S(r, v) Stirling number of the second kind GF (q) Galois field of q elements N set of coordinates n code length w code weight d code minimum distance e radius C code J(n, w) Johnson graph Φe(n, w) size of a sphere of radius e in J(n, w) t (n,w,λ) t-design over n elements and blocks of size w S−(t, w, n) Steiner system over n elements and blocks of size w ϕ code strength S(t1, t2,w1,w2, n1, n2) doubly Steiner system Φe(n1, n2,w1,w2) size of a sphere of radius e in doubly constant code 3 Chapter 1 Introduction Codes which attain the sphere packing bound are called perfect code. Perfect codes al- ways draw the attention of coding theoreticians and mathematicians. The most important metrics in coding theory on which perfect codes are defined are the Hamming metric and the Johnson metric. In the Hamming metric, all perfect codes over finite fields are known [1]. They exist for only a small number of parameters, while for other parameters their non-existence was proved [2, 3, 4, 1]. The nonexistence proof is based on Lloyd’s polynomials. No nontrivial perfect code is known over other alphabets and for most parameters it was proved that they do not exist [5]. As for the Johnson metric, it was conjectured by Delsarte [6] in 1973 that there are no nontrivial perfect codes. Many attempts were made during the last 35 years to prove this conjecture. These attempts used Lloyd polynomials, anticodes, designs and num- ber theory. However, the previous research yielded only partial results and the general nonexistence is yet to be proved. Perfect codes in the Johnson metric have a strong connection to constant weight codes. Constant weight codes play an important role in various areas of coding theory. One of their applications is in obtaining lower and upper bounds on the sizes of unrestricted codes for given length and minimum Hamming distance [7, 1]. In the same way as constant weight codes are used for obtaining bounds on the sizes of unrestricted codes, doubly constant weight codes play an important role in obtaining bounds on the sizes of constant weight codes [8]. A natural question is whether there exist perfect doubly constant weight codes. 4 1.1 Definitions A binary unrestricted code of length n is the set of binary words of length n. The weight of a word is the number of ones in the word. A constant weight code of length n and weight w is a binary code whose codewords have constant weight w. A doubly constant weight code of length n and weight w is a constant weight code of length n and weight w, with w1 ones in the first n1 positions and w2 ones in the last n2 positions, where n = n1 + n2 and w = w1 + w2 . The Hamming distance (or H-distance in short) between two words of the same length n is the number of coordinates in which they differ. If we define the distance between two words, x and y of the same weight w and the same length n, as half their H-distance, we obtain a new metric which is called the Johnson metric and the distance is called the Johnson distance (or J-distance in short). Let A(n, d) denote the maximum number of codewords in a binary code of length n and minimum H-distance d Let A(n,d,w) denote the maximum number of codewords in a constant weight code of length n, weight w and minimum H-distance d. A (w1, n1,w2, n2,d) code is a doubly constant weight code with w1 ones in the first n1 positions and w2 ones in the last n2 positions, and minimum J-distance d . Let T (w1, n1,w2, n2, δ) denote the maximum number of codewords in a (w1, n1,w2, n2,d) code, where δ =2d is a H-distance. 1.1.1 Block designs There is a tight connection between constant weight codes and block designs. In the next chapters we will use the following terminology and properties of block designs. Definition. Let t,n,w,λ be integers with n>w t and λ > 0. Let N be an n-set (i.e.
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