New Bounds for Linear Codes of Covering Radius 2

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.

Designs, Codes and Cryptography, 2010

The shortest possible length of a q-ary linear code of covering radius R and codimension r is called the length function and is denoted by q (r, R). Constructions of codes with covering radius 3 are here developed, which improve best known upper bounds on q (r, 3). General constructions are given and upper bounds on q (r, 3) for q = 3, 4, 5, 7 and r ≤ 24 are tabulated.

The Electronic Journal of Combinatorics, 2009

Let $K_q(n,R)$ denote the minimal cardinality of a $q$-ary code of length $n$ and covering radius $R$. Recently the authors gave a new proof of a classical lower bound of Rodemich on $K_q(n,n-2)$ by the use of partition matrices and their transversals. In this paper we show that, in contrast to Rodemich&#39;s original proof, the method generalizes to lower-bound $K_q(n,n-k)$ for any $k&gt;2$. The approach is best-understood in terms of a game where a winning strategy for one of the players implies the non-existence of a code. This proves to be by far the most efficient method presently known to lower-bound $K_q(n,R)$ for large $R$ (i.e. small $k$). One instance: the trivial sphere-covering bound $K_{12}(7,3)\geq 729$, the previously best bound $K_{12}(7,3)\geq 732$ and the new bound $K_{12}(7,3)\geq 878$.

2018

The length function ℓ_q(r,R) is the smallest length of a q -ary linear code of codimension r and covering radius R. In this work we obtain new constructive upper bounds on ℓ_q(r,R) for all R&gt;4, r=tR, t&gt;2, and also for all even R&gt;2, r=tR+R/2, t&gt;1. The new bounds are provided by infinite families of new covering codes with fixed R and increasing codimension. The new bounds improve upon the known ones. We propose a general regular construction (called &quot;Line+Ovals&quot;) of a minimal ρ-saturating ((ρ+1)q+1)-set in the projective space PG(2ρ+1,q) for all ρ&gt;0. Such a set corresponds to an [Rq+1,Rq+1-2R,3]_qR locally optimal^1 code of covering radius R=ρ+1. Basing on combinatorial properties of these codes regarding to spherical capsules^1, we give constructions for code codimension lifting and obtain infinite families of new surface-covering^1 codes with codimension r=tR, t&gt;2. In addition, we obtain new 1-saturating sets in the projective plane PG(2,q^2) and, basing...

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.

Ieee Transactions on Information Theory, 2004

Infinite families of linear codes with covering radius = 2, 3 and codimension + 1 are constructed on the base of starting codes with codimension 3 and 4. Parity-check matrices of the starting codes are treated as saturating sets in projective geometry that are obtained by computer search using projective properties of objects. Upper bounds on the length function and on the smallest sizes of saturating sets are given.

IEEE Transactions on Information Theory, 2004

Infinite families of linear codes with covering radius = 2, 3 and codimension + 1 are constructed on the base of starting codes with codimension 3 and 4. Parity-check matrices of the starting codes are treated as saturating sets in projective geometry that are obtained by computer search using projective properties of objects. Upper bounds on the length function and on the smallest sizes of saturating sets are given.

Ieee Transactions on Information Theory, 1998

k01 j=0 dd=q j e = N d + A + k 0 I d 0 2; so k n 0 N d + I d 0 A + 2: Comparing this inequality with Theorem 4, we completed the proof. In the following, we give two examples. Example 1: q = 2, d = 5. If n > 12, the new bound is tighter than the Griesmer bound. For instance, the Griesmer bound cannot prove the nonexistence [13; 6; 5], but the new bound can. Example 2: q = 3, d = 9. If n > 13, the new bound is tighter than the Griesmer bound. For instance, the Griesmer bound cannot prove the nonexistence [14; 4; 9], but the new bound can. IV. CONCLUSION In this correspondence, we gave some bounds for [n; k; d] codes with the method of coset partitions. These bounds are new in type and are derived from the basic properties of the parity-check matrix. Comparing these bounds with tables of the best codes known, it can be shown that these bounds are not strong for arbitrary n and d although, indeed, Theorem 4 improves on the Griesmer bound for a large range of code parameters. Comparing with [3, Table I] in the binary case, we list some results in the following. Case 1: d = 3. In this case, Theorem 1 shows that k bn 0 log 2 (n + 1)c: The equality holds for Hamming [n = 2 r 0 1; k = n 0 r; d = 3] codes and also for shortened Hamming [n = 2 r 010m; k = n 0r 0m; d = 3] codes, where 0 < m < n 0 r. This shows that Theorem 1 is tight for d = 3. This also shows that the Hamming codes and shortened Hamming codes are optimal linear block codes for a given codelength and minimum distance d = 3, a fact known earlier in coding theory. Case 2: d = 5. We compare Theorem 2 with [3, Table I ]. In this case, Theorem 2 shows that k bn 0 4 0 log 2 (n 0 4)c. The results obtained are listed in Table I. Case 3: d = 7. In this case, Theorem 3 can be written as k bn 0 7 0 log 2 (n 0 7)c: The results obtained are listed in Table II. Case 4: d = 9. In this case, Theorem 4 can be written as k bn 0 12 0 log 2 (n 0 13)c:

Advances in Mathematics of Communications

The length function \begin{document}$ \ell_q(r,R) $\end{document} is the smallest length of a \begin{document}$ q $\end{document}-ary linear code with codimension (redundancy) \begin{document}$ r $\end{document} and covering radius \begin{document}$ R $\end{document}. In this work, new upper bounds on \begin{document}$ \ell_q(tR+1,R) $\end{document} are obtained in the following forms: \begin{document}$ \begin{equation*} \begin{split} &(a)\; \ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &\phantom{(a)\; } q\;{\rm{ is \;an\; arbitrary \;prime\; power}},\; c{\rm{ \;is\; independent \;of\; }}q. \end{split} \end{equation*} $\end{document} \begin{document}$ \begin{equation*} \begin{split} &(b)\; \ell_q(r,R)&lt; 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},\; R\ge3,\; r = tR+1,\; t\ge1,\\ &\phantom{(b)\; } q\;{\rm{ is \;an\; arbitrary\; prime \;power}},\; q\;{\rm{ is \;large\; enough}}. \end{split} \end{equation*} $\end{document} In the literature, for \begin{doc...

IEEE Transactions on Information Theory, 1986

A number of upper and lower bounds are obtained for K( n, R), the minimal number of codewords in any binary code of length n and covering radius R. Several new constructions are used to derive the upper bounds, including an amalgamated direct sum construction for nonlinear codes. This construction works best when applied to normal codes, and we give some new and stronger conditions which imply that a linear code is normal. An upper bound is given for the density of a covering code over any alphabet, and it is shown that K(n + 2,R + 1) 5 K(n, R) holds for sufficiently large n.