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New
example [2], [3]. In Sections II and III a new [47; 15; 16] code is presented that reaches the upper bound on code distance for n = 47 and k = 15 and its construction is explained. Finally some conclusions are drawn.
Linear Binary Block Code
Martin Janoˇsov, Martin Husák, Peter Farkaˇs, Member, IEEE, and Ana Garcia Armada, Member, IEEE
II. NEW CODE Abstract—A new [47; 15; 16] linear binary block code and its weight spectrum is presented. The code is better than the previously known [47; 15; 15] code and it reaches the upper bound on code distance for the codeword length 47 and dimension 15. Index Terms—Code distance, computerized search, error control code, linear binary block code, upper bound, weight spectrum.
I. INTRODUCTION
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Linear binary [n; k; d]-code is defined as a k -dimensional linear subspace of n-dimensional vector space over finite field GF(2), where 2 it is denoted d denotes its code distance [1]. If the vector as a codeword of the code . Code distance d is equal to minimum Hamming distance (number of different coordinates) between any two codewords of the code. One of the central problems in coding theory is to find the largest value of d for given values of n and k denoted dmax (n; k ). The lower bounds Lbfdmax (n; k )g and the upper bounds U bfdmax (n; k )g on dmax (n; k ) are collected in different tables, for
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c
G
The new [47; 15; 16]-code is described by generator matrix shown in (1) and its weight distribution using Aj for the number of codewords of Hamming weight j . (The Hamming weight of the vector w(c) is the number of nonzero coordinates in c). The weight distribution of the code is: A0 = 1, A16 = 1755, A20 = 8683, A24 = 15090, A28 = 6390, A32 = 834, A36 = 15. (See (1) at the bottom of the page.) The [47; 15; 16]-code improves the lower bound in [3], which before was determined by [47; 15; 15]-code [4]. It also reaches the upper bound on code distance for codeword length 47 and dimension 15. In online version of [5] of table [2] which was published also in [6] it is given that: ” U bfdmax (47; 15)g = 16 follows by a one-step Griesmer bound from U bfdmax (30; 14)g = 8, which follows by a one-step Griesmer bound from U bfdmax (21; 13)g = 4, which follows by a one-step Griesmer bound from U bfdmax (16; 12)g = 2, which is found by construction B via deleting at most eight coordinates of a word in the dual.”
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III. CONSTRUCTION The new code was originally found via computerized search using algorithm B described in [4] with a starting matrix found heuristically. Later the code was analyzed using MAGMA [7] and it was pointed out that the code could be also obtained using construction X [1] from the following codes: 1) [2; 1; 2] cyclic linear code over GF(2) (Repetition Code of length 2); 2) [45; 14; 16] quasi-cyclic code of degree 3 over GF(2) with generating polynomials:
M. Janoˇsov, M. Husák, and P. Farkaˇs are with the Slovak University of Technology, 812 19 Bratislava, Slovakia (e-mail:
[email protected];
[email protected];
[email protected]). A. G. Armada is with the Department of Signal Theory and Communications, Universidad Carlos III de Madrid, c/Butarque 15, 289 11 Leganés, Spain (e-mail:
[email protected]).
x
13
+x
4
7 6 2 +x +x +x 14 9 5 13 12 4 2 10 x + x + x + x + x + x + x + x + x + 1; x
14
10000000000001100001110011110100101001111001011 01000000000001000111111100111001010000101000000 00100000000000100011111110011100101000010100000 00010000000001100110111010001101010000011110111 00001000000001000100011000000111101100011011100 00000100000000100100111010110110011011100000000
G=
00000010000001100101011000011000001001100100111 00000001000001000011011111111010101101001011000 00000000100000100001101111111101010110100101100 00000000010001100111110010111111101111000110011 00000000001001000100111100011100110011110111100 00000000000100100100101000111011110100010110000 00000000000011100011100111101001010011110010011 00000000000000010110110110110101101101101101100 00000000000000001101101101101111011011011011000
1
:
(1)
3) [45; 15; 14] quasi-cyclic code of degree 3 GF(2) with generating polynomials:
+ x13 + 1 9 6 5 11 4 12 10 x +x +x +x +x +x +x +x 13 8 7 4 10 2 9 x +x +x +x +x +x +x : x
14
14
IV. CONCLUSION The New [47; 15; 16] code was found using computerized search and later its analysis using MAGMA revealed another procedure for its construction, which is also described in this paper. The lower and upper bound on dmax (47; 15) is now equal to 16. The actual entries in [3] for n = 48 and dimension k = 16 suggest that the code [48; 16; 16] can exist. Therefore we made some search using the algorithm B from [3] to find that code as well, but without success until now. We believe that additional attempts in future with different methods or different starting conditions could be more successful. ACKNOWLEDGMENT The authors would like to express their thanks to Markus Grassl for the analysis of the code using MAGMA, which resulted in the second construction presented in this correspondence.
REFERENCES [1] F. J. McWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland Mathematical Library, 1977. [2] A. E. Brouwer and T. Verhoeff, An Updated Table of Minimum-Distance Bounds for Binary Linear Codes. Eindhoven, Germany: University of Eindhoven. [3] [Online]. Available: http://www.codetables.de/ [4] P. Farkaˇs and K. Brühl, “Three best binary linear block codes of minimum distance fifteen,” IEEE Trans. Inf. Theory, vol. 40, no. 3, pp. 949–951, May 1994. [5] [Online]. Available: http://www.win.tue.nl/~aeb/voorlincod.html [6] V. S. Pless and W. C. Huffman, Handbook of Coding Theory. Amsterdam, The Netherlands: Elsevier, 1998. [7] [Online]. Available: http://magma.maths.usyd.edu.au/
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