Tables, bounds and graphics of short linear codes with covering radius 3 and codimension 4 and 5
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The length function ℓ_q(r,R) is the smallest length of a q -ary linear code of covering radius R and codimension r. In this work, by computer search in wide regions of q, we obtained short [n,n-4,5]_q3 quasiperfect MDS codes and [n,n-5,5]_q3 quasiperfect Almost MDS codes with covering radius R=3. The new codes imply the following upper bounds: &ℓ_q(4,3)<2.8√(q q) for 8< q<3323 and q=3511,3761,4001; &ℓ_q(5,3)<3√(q^2 q) for 5< q<563. For r≠ 3t and q≠ (q^')^3, the new bounds have the form ℓ_q(r,3)< c√( q)· q^(r-3)/3, c is a universal constant, r=4,5. As far as it is known to the authors, such bounds have not been previously described in the literature. In computer search, we use the leximatrix algorithm to obtain parity check matrices of codes. The algorithm is a version of the recursive g-parity check algorithm for greedy codes.
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