Linear codes with covering radius 3

arXiv:1712.07078v5 [cs.IT], 2019

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.

IEEE Transactions on Information Theory, 1999

New constructions of linear nonbinary codes with covering radius R = 2 are proposed. They are in part modifications of earlier constructions by the author and in part are new. Using a starting code with R = 2 as a "seed" these constructions yield an infinite family of codes with the same covering radius. New infinite families of codes with R = 2 are obtained for all alphabets of size q 4 and all codimensions r 3 with the help of the constructions described. The parameters obtained are better than those of known codes. New estimates for some partition parameters in earlier known constructions are used to design new code families. Complete caps and other saturated sets of points in projective geometry are applied as starting codes. A table of new upper bounds on the length function for q = 4; 5; 7; R = 2; and r 24 is included.

Designs, Codes and Cryptography, 2007

We determine the minimum length n q (k, d) for some linear codes with k ≥ 5 and q ≥ 3. We prove that n q (k, d) = g q (k, d) + 1 for q k−1 − 2q k−1 2 −q + 1 ≤ d ≤ q k−1 − 2q k−1 2 when k is odd, for q k−1 − q k 2 − q k 2 −1 − q + 1 ≤ d ≤ q k−1 − q k 2 − q k 2 −1 when k is even, and for 2q k−1 − 2q k−2 − q 2 − q + 1 ≤ d ≤ 2q k−1 − 2q k−2 − q 2 .

2017

The length function \(\ell _q(r,R)\) is the smallest length of a q-ary linear code of covering radius R and codimension r. New upper bounds on \(\ell _q(r,2)\) are obtained for odd \(r\ge 3\). In particular, using the one-to-one correspondence between linear codes of covering radius 2 and saturating sets in the projective planes over finite fields, we prove that $$\begin{aligned} \ell _q(3,2)\le \sqrt{q(3\ln q+\ln \ln q)}+\sqrt{\frac{q}{3\ln q}}+3 \end{aligned}$$ and then obtain estimations of \(\ell _q(r,2)\) for all odd \(r\ge 5\). The new upper bounds are smaller than the previously known ones. Also, the new bounds hold for all q, not necessary large, whereas the previously best known estimations are proved only for q large enough.

IEEE Transactions on Information Theory, 2000

Finite Fields and Their Applications, 2000

A new quaternary linear code of length 19, codimension 5, and covering radius 2 is found in a computer search using tabu search, a local search heuristic. Starting from this code, which has some useful partitioning properties, di!erent lengthening constructions are applied to get an in"nite family of new, record-breaking quaternary codes of covering radius 2 and odd codimension. An algebraic construction of covering codes over alphabets of even characteristic is also given.

2017

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring R = F q + vF q + v 2 F q , where v 3 = v, for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over F q and extend these to codes over R.

be the smallest integer n for which there exists a linear code of length n, dimension IC, and minimum distance d, over a field of q elements. In this correspondence we determine n5 (4, d ) for all but 22 values of d. Index Terms-Optimal q-ary linear codes, minimum-length bounds. Publisher Item Identifier S 0018-9448(97)00108-9.

IEEE Transactions on Information Theory, 2001

Let [ ] denote a linear code over with length , codimension , and covering radius . We use a modification of constructions of [2 +1 2 3] 2 and [3 +1 3 5] 3 codes ( 5) to produce infinite families of good codes with covering radius 2 and 3 and codimension .

Let an $[n,k,d]_q$ code be a linear code of length $n$, dimension $k$ and minimum Hamming distance $d$ over $GF(q)$. One of the most important problems in coding theory is to construct codes with optimal minimum distances. In this paper 22 new ternary linear codes are presented. Two of them are optimal. All new codes improve the respective lower bounds in [11].