| Linear Programming Bounds for Codes of Small Size (2007) | |||||||||||||||
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| Combining linear programming approach with the Plotkin-Johnson argument for constant weight codes, we derive upper bounds on the size of codes of length n and minimum distance d = (n 0 j)=2, 0 ! j ! n 1=3 . For j = o(n 1=3 ) these bounds practically coincide (are slightly better) with the Tietavainen bound. For fixed j and j proportional to n 1=3 , j ! n 1=3 0 (2=9) ln n, it improves on the earlier known results. Keywords: Upper bounds, Plotkin bound, Tietavainen bound, McEliece bound. 1 Introduction Let C be a binary (n; M; d) code of length n, size M , and minimum distance d = (n0j)=2. Let B = (B 0 ; B 1 ; . . . ; B n ) stand for its distance distribution. For every c 2 C define the relative weight distribution A(c) = (A 0 (c); . . . ; A n (c)) w.r.t. c, where A i (c) denotes the number of codewords of C being at distance i from c. So, B i = 1 M X c2C A i (c): The MacWilliams transform of B is B 0 k = 1 M n X i=0 B i P k (i); and in every code B 0 = B 0 0 = 1,... | |||||||||||||||
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