International Mathematical Forum, Vol. 6, 2011, no. 4, Manjusri Basu

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1 International Mathematical Forum, Vol 6, 011, no 4, Square Designs on New Binary ( 3n 1, 3 n ) Codes Manjusri Basu Department of Mathematics University of Kalyani Kalyani, WB, India, Pin manjusri basu@yahoocom Satya Bagchi Department of Mathematics National Institute of Technology, Durgapur Burdwan, WB, India, Pin satya5050@gmailcom Debabrata Kumar Ghosh Department of Mathematics University of Kalyani Kalyani, WB, India, Pin debabrataghsh03@gmailcom Abstract C is an (n, M, d) q code over F q of length n with M codewords and minimum distance d The code C can be either linear or nonlinear A t (v,k, λ) design D is a set X of v points together with a collection of k subsets of X (called block) such that every t subset of X is contained in exactly λ blocks In this paper a new binary code ( 3n 1, 3 n ), n 3 is developed Some properties of this code are stated It is also shown that this code holds ( 3n 1, 3 n 1, 3 n ) design Keywords: Linear code, constant weight code, support, complement code, t-design 1 Introduction Let Fq n denote the linear space of all n-tuples over the finite field F q = GF (q) An (n, M) codec over F q is a subset of Fq n of size M If C is a k-dimensional

2 186 M Basu, S Bagchi and D K Ghosh subspace of F n q, C is called an [n, k] linear code over F q The field F is very special in coding theory, and codes over F are called binary codes Codes over F 3 are called ternary codes, and codes over F 4 are called quaternary codes and so on Thus codes over F q are called q ary codes It is natural to expect that for some integers n and d there exists an (n, M, d) q nonlinear q-ary code of length n with minimum distance d whose number of codewords M is greater than the number of codewords of any linear q-ary code of length n and minimum distance d In ternary q =3,C is an (n, M, d) 3 nonlinear code [6] Due to the rapid growth of applications of design theory in modern science, various type of new codes holding with designs are coming up in the literature [1,,4,5,7,10] In this paper, a systematic new binary code ( 3n 1, 3 n ),n 3is defined The different properties of this code are stated and proved It is also shown that this code holds ( 3n 1, 3 n 1, 3 n ) design Definitions Definition 1 Complement of a Binary code[3]: The complement of a binary code C is the binary code C + 1, where 1 is the all one codeword; so the complement of C is the code obtained from C by replacing 1 by 0 and 0 by 1 Definition Support[8]: Let c be a binary codeword of length n The set of positions in which c has nonzero entries is called support of c Definition 3 Design[9]: Let C be a binary code of length n Let S k be the set of codewords in C of weight k We say that S k holds a t (n, k, λ) design if the supports of codewords in S k form the blocks of a t (n, k, λ) design, if for any t-set T {1,,,n} there are exactly λ codewords of weight k in C with 1 s in the positions given by T A t (v, k, 1) design is defined as a Steiner System and denoted by S(t, k, v) 3 The new code C n (n 3) Let A n (n 3) be a matrix of order n (3 n 1) in F 3 The columns of A n are all ternary codewords of length n except all zero codeword 0 The matrix G n of order n 3n 1 in F 3 is obtained after deleting the linearly dependent columns of A n Now the matrix P n of order (3 n 1) 3n 1 is generated by G n in F 3 except 0 codeword In P n, 1 is taken in place of So P n has each row twice After

3 Square designs on new binary codes 187 deleting the same row of P n, the matrix C n of order 3n 1 Hence the new code C n is constructed For example: When n =3, A 3 = 3n 1 is obtained After deleting the linearly dependent columns of A 3, G 3 = After deleting 0 codeword from the code generated by G 3, P 3 = Replacing by 1 and deleting the same rows of P 3, the new code

4 188 M Basu, S Bagchi and D K Ghosh C 3 = The complement code C c 3 of C 3 is C3 c = Similarly C n and Cn c can be constructed for all values of n 4 Properties Property 41 The code C n is a nonlinear code Property 4 The code C n = C n, where C n is the square matrix of order 3 n 1 obtained by re-arranging the columns of C t n, the transpose of C n Property 43 Distance of any two different codewords of C n is 3 n Proof: Any two columns of the matrix P n are linearly independent In the matrix P n, only 3 n 1 rows of any two columns have consecutive 0 s, 43 n rows of any two columns have consecutive nonzero and the rest 43 n =

5 Square designs on new binary codes 189 (3 n 1) (3 n 1) 43 n rows of any two columns have one zero and one nonzero Now after replacing by 1 and deleting 3n 1 identical rows of P n, C n is obtained Thus in C n, any two columns have consecutive 0 s in 3n 1 rows, consecutive 1 s in 3 n (= 43n ) rows and one 0 and one 1 in 3 n rows Hence by property, distance of any two different codewords of C n is 3 n Property 44 Weight of each codeword of the code C n is 3 n 1 Proof: P n including 0 codeword is a linear code So in each column of P n, the numbers of 0 s, 1 s and s are 3 n 1 1, 3 n 1 and 3 n 1 respectively After replacing by 1, the numbers of 0 s and 1 s are 3 n 1 1 and 3 n 1 respectively in each column of P n After deleting the same row of P n, the numbers of 0 s and 1 s are reduced to 3n 1 1 and 3 n 1 respectively Hence by property, the weight of each codeword of C n is 3 n 1 Property 5 and property 6 follow from property 4: Property 45 Weight of the every codeword c + 1 (c C n ) of the complement codeword C c n of C n is 3n 1 1 Property 46 The code C n is a 3 n -error correcting code Hence the new code C n ( 3n 1, 3 n ) is defined for n 3 5 Main Results Theorem 51 The code C n ( 3n 1, 3 n ) holds ( 3n 1, 3 n 1, 3 n ) design, n 3 Proof: The length and weight of each codeword of C n are 3n 1 and 3 n 1 respectively Also by property 3, in any two columns of C n there are 3 n consecutive 1 s Hence the code C n ( 3n 1, 3 n ) holds ( 3n 1, 3 n 1, 3 n ) design, n 3 Theorem 5 The code C n ( 3n 1, 3 n ) does not hold 3-design, n 3 Proof: Considering 1 st, nd and 3 rd columns of C 3, there are three consecutive 1 s But considering 1 st, nd and 5 th columns of C 3, there are four consecutive 1 s So the code C 3 does not hold 3 design Hence the code C n does not hold 3 design

6 190 M Basu, S Bagchi and D K Ghosh Theorem 53 The complement code Cn c( 3n 1 3 n 1) design, n 3, 3 n ) hold ( 3n 1, 3n 1 1, Proof: The length and weight of the complement code Cn c of C n are 3n 1 respectively By property 3 any two columns of the matrix C c n have and 3n n 1 consecutive 1 s Hence the theorem For n =3, C c 3 holds Steiner System, ei C c 3 holds S(, 4, 13) Theorem 54 The complement code Cn c( 3n 1, 3 n ) does not hold 3-design, n 3 Proof: Considering 10 th, 11 th and 1 th columns of C3 c, there is no consecutive 1 s But considering 11 th, 1 th and 13 th columns of C3 c, there is a consecutive 1 s So the code C3 c n 3 does not hold 3 design Hence the code Cn c does not hold 3 design 6 Conclusion There are close relations between design theory and coding theory In recent years, design theory has grown up tremendously with computer science Also it has become necessary for interdisciplinary research with pure and applied mathematics groups, industrial groups etc References [1] M Basu, Md M Rahaman and S Bagchi, On a new code, [ n,n, n 1 ], Discrete Applied Mathematics, 175 (009), [] I Bouyukliev, V Fack, and J Winne, -(31,15,7), -(35,17,8) and - (36,15,6) designs with automorphisms of odd prime order, and their related Hadamard matrices and codes, Design, Codes and Cryptography, 51 (009), [3] W C Huffman and V Pless, Fundamentals of Error-Correcting Codes, Cambridge, 004 [4] D R Hughes and F C Piper, Design Theory, Cambridge University Press, Cambridge, 1985 [5] Y J Ionin and H Kharaghani, A Recursive Construction for New Symmetric Designs, Designs, Codes and Cryptography, 35 (005),

7 Square designs on new binary codes 191 [6] S Ling and C Xing, A first course of coding theory, National University of Singapore, 004 [7] P R J Ostergard, A -(, 8, 4) Design Cannot Have a -(10, 4, 4) Subdesign, Designs, Codes and Cryptography, 7 (00), [8] V S Pless and WC Huffman, Handbook of coding theory Vol II, North Holland, 1998 [9] S Roman, Coding and Information Theory, Springer, 1991 [10] D J Shin, P V Kumar and T Helleseth, 3-Designs from the Z 4 -Goethals Codes via a New Kloosterman Sum Identity, Designs, Codes and Cryptography, 8 (003), Received: July, 010

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