Tables, bounds and graphics of short linear codes with covering radius 3 and codimension 4 and 5

The length function $\ell_q(r,R)$ is the smallest length of a $q$-ary linear code of codimension (redundancy) $r$ and covering radius $R$. The $d$-length function $\ell_q(r,R,d)$ is the smallest length of a $q$-ary linear code with codimension $r$, covering radius $R$, and minimum distance $d$. By computer search in wide regions of $q$, we obtained following short codes of covering radius $R=3$: $[n,n-4,5]_q3$ quasi-perfect MDS codes, $[n,n-5,5]_q3$ quasi-perfect Almost MDS codes, and $[n,n-5,3]_q3$ codes. In computer search, we use the step-by-step leximatrix and inverse leximatrix algorithms to obtain parity check matrices of codes. The new codes imply the following new upper bounds (called lexi-bounds) on the length and $d$-length functions: $$\ell_q(4,3)\le\ell_q(4,3,5)<2.8\sqrt[3]{\ln q}\cdot q^{(4-3)/3}=2.8\sqrt[3]{\ln q}\cdot\sqrt[3]{q}=2.8\sqrt[3]{q\ln q}~\text{for}~11\le q\le7057;$$ $$\ell_q(5,3)\le\ell_q(5,3,5)<3\sqrt[3]{\ln q}\cdot q^{(5-3)/3}=3\sqrt[3]{\ln q}\cdot\sqrt[3]{q^2}=3\sqrt[3]{q^2\ln q}~~\text{ for }~37\le q\le839.$$ Moreover, we improve the lexi-bounds, applying randomized greedy algorithms, and show that $$\ell_q(4,3)\le \ell_q(4,3,5)< 2.61\sqrt[3]{q\ln q}~\text{ if }~13\le q\le4373;$$ $$\ell_q(4,3)\le \ell_q(4,3,5)< 2.65\sqrt[3]{q\ln q}~\text{ if }~4373<q\le7057;$$ $$\ell_q(5,3)<2.785\sqrt[3]{q^2\ln q}~\text{ if }~11\le q\le401;$$ $$\ell_q(5,3)\le\ell_q(5,3,5)<2.884\sqrt[3]{q^2\ln q}~\text{ if }~401<q\le839.$$ The codes, obtained in this paper by leximatrix and inverse leximatrix algorithms, provide new upper bounds (called density lexi-bounds) on the smallest covering density $\mu_q(r,R)$ of a $q$-ary linear code of codimension $r$ and covering radius $R$: $$\mu_q(4,3)<3.3\cdot\ln q~~\text{ for }~11\le q\le7057;$$ $$\mu_q(5,3)<4.2\cdot\ln q~~\text{ for }~37\le q\le839.$$