Complexity of Computation by R. Karp

By R. Karp

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The Hadamard code thus has very poor rate since it maps 3 When m ≥ q, in general there is no simple closed form for the dimension, and the relative distance is at least q − m/q . 22 2 Preliminaries and Monograph Structure symbols over Fq into q symbols. But it has very good distance properties — its relative distance equals (1 − 1/q), and in fact every non-zero codeword has Hamming weight equal to (q − q −1 ). Despite its poor rate, its highly structured distance properties makes it an attractive code for use at the inner level in certain concatenation schemes.

Vm all lie in RN , they must be linearly dependent. Let S ⊆ [m] be a non-empty set of minimum size for which a relation of the form i∈S ai vi = 0 holds with each ai = 0. We claim that the ai ’s must all be positive or all be negative. Indeed, if not, by collecting terms with positive ai ’s on one side and those with negative ai ’s on the other, we will have an equation of the form i∈T + ai vi = j∈T − bj vj = w (for some vector w) where T + and T − are disjoint non-empty sets with T + ∪ T − = S, and all ai , bj > 0.

7. Let C ⊆ [q]n be a code of blocklength n and minimum distance d. Let {wi,j : 1 ≤ i ≤ n; 1 ≤ j ≤ q} be an arbitrary set of non-negative real weights. (i) The number of codewords C ∈ C that satisfy n wi,Ci > (n − d) i=1 is at most (nq − 1). 17) i,j is at most L. 7 above can also be worked out for the case when the diﬀerent codeword positions have diﬀerent contributions towards the minimum distance. 1 of the book. 10. 5 Notes The quantity A(n, d, w) for constant-weight binary codes has a rich history and has been studied for almost four decades, and its study remains one of the most basic questions in coding theory.