Optimal quaternary linear codes of dimension five

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.

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 .

Acta Scientiarum. …, 1999

IEEE Transactions on Information Theory, 2000

European Journal of Combinatorics, 2004

We determine the parameters of the optimal additive quaternary codes of length at most 12 over Z 2 × Z 2 . Equivalently, we determine how many lines one can pick in a binary projective space such that any t are independent. Or again, how many lines one can pick in a binary projective space such that no hyperplane contains more than m of them.

Discrete Mathematics, 1990

A record binary code of length 45, dimension 13 and minimal distance 16 is constructed in several ways: as an Abelian code (an ideal in the regular representation of C, x C,,), from a [15,6,8] cyclic code over GF(4), and from the la +x (b +x 1 c +x1 construction. Its automorphism group has order 360, and its even subcode is a [45,12,16] code with only four nonzero weights.

q code be a linear code of length n, dimension k and Hamming minimum distance d over GF(q). In this paper record-breaking codes with parameters [30, 10, 15]5, [

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.

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].

Discrete Mathematics, 1994

A central problem in coding theory is that of finding the smallest length for which there exists a linear code of dimension k and minimum distance d, over a field of ~7 elements, We consider here the problem for quaternary codes (q=4), solving the problem for k< 3 for all values of d, and for k=4 for all but ten values of d.