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The covering radius of C is deï¬ned as r(C)=max xâF min yâC+x wt(x), where wt(Â·) denotes the Hamming weight. Bounding the covering radius of codes is one of the main extremal problems of coding theory. Let C ={x â F: (x,c)=0 âc âC} be the dual code of C and (A i,0 â¤i â¤n) be its weight distribution, where A i =|{x âC: wt(x)=i}|and (x,c)denotes the dot product.
the most likely codeword x to have been transmitted is the one for which the fewest errors need to have. occurred in order for y to have been received. The Hamming distance between two vectors, denoted dist(x;y), is the number of i such that xi 6= yi. The weight of a vector, wt(x), is the number of nonzero xi. Obviously, dist(x;y) = wt(x y). Maximum likelihood decoding computes the distance between the received word y andeach codeword x, and decodes