Papers by Stefano Marcugini
arXiv:1712.07078v5 [cs.IT], 2019
The length function q (r, R) is the smallest length of a q-ary linear code of codimension (redund... more The length function q (r, R) is the smallest length of a q-ary linear code of codimension (redundancy) r and covering radius R. The d-length function q (r, R, d) is the smallest length of a q-ary linear code with codimension (redundancy) r, covering radius R, and minimum distance d. By computer search in wide regions of q, we obtained following short codes of covering radius R = 3: [n, n − 4, 5] q 3 quasi-perfect MDS codes, [n, n − 5, 5] q 3 quasi-perfect Almost MDS codes, and [n, n − 5, 3] q 3 codes. In computer search, we use the step-by-step lexi-matrix and inverse leximatrix algorithms to obtain parity check matrices of codes. These algorithms are versions of the recursive g-parity check matrix algorithm for greedy codes. The new codes imply the following new upper bounds (called lexi-bounds) on the length function and the d-length function: q (4, 3) ≤ q (4, 3, 5) < 2.8 3 ln q · q (4−3)/3 = 2.8 3 ln q · 3 √ q = 2.8 3 q ln q for 11 ≤ q ≤ 6607; q (5, 3) ≤ q (5, 3, 5) < 3 3 ln q · q (5−3)/3 = 3 3 ln q · 3 q 2 = 3 3 q 2 ln q for 37 ≤ q ≤ 839. Moreover, we improve the lexi-bounds, applying randomized greedy algorithms, and show that q (4, 3) ≤ q (4, 3, 5) < 2.61 3 q ln q if 13 ≤ q ≤ 4373; q (4, 3) ≤ q (4, 3, 5) < 2.65 3 q ln q if 4373 < q ≤ 6607; q (5, 3) < 2.785 3 q 2 ln q if 11 ≤ q ≤ 401; q (5, 3) ≤ q (5, 3, 5) < 2.884 3 q 2 ln q if 401 < q ≤ 839. The general form of the new bounds is q (r, 3) < c 3 ln q · q (r−3)/3 , c is a constant independent of q, r = 4, 5 = 3t. The codes, obtained in this paper by leximatrix and inverse leximatrix algorithms, provide the following new upper bounds (called density lexi-bounds) on the smallest covering density µ q (r, R) of a q-ary linear code of codimension r and covering radius R: µ q (4, 3) < 3.3 · ln q for 11 ≤ q ≤ 6607; µ q (5, 3) < 4.2 · ln q for 37 ≤ q ≤ 839. In the general form, we have µ q (r, 3) < c µ · ln q, c µ is a constant independent of q, r = 4, 5. The new bounds on the length function, the d-length function and covering density hold for the field basis q of an arbitrary structure, including q = (q) 3 where q is a prime power.
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Papers by Stefano Marcugini