Bounds for Binary Linear Locally Repairable Codes via a Sphere-Packing Approach

For locally repairable codes (LRCs), Cadambe and Mazumdar derived the first field-dependent parameter bound, known as the C-M bound. However, the C-M bound depends on an undetermined parameter <inline-formula> <tex-math notation="LaTeX">$ {k}^{( {q})}_{{\textbf {opt}}}( {n}, {d})$ </tex-math></inline-formula>. In this paper, a sphere-packing approach is developed for upper bounding the parameter <inline-formula> <tex-math notation="LaTeX">$ {k}$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$[ {n}, {k}, {d}]$ </tex-math></inline-formula> linear LRCs with locality <inline-formula> <tex-math notation="LaTeX">$ {r}$ </tex-math></inline-formula>. When restricted to the binary field, three upper bounds (i.e., <monospace>Bound A</monospace>, <monospace>Bound B</monospace>, and <monospace>Bound C</monospace>) are derived in an explicit form. More specifically, <monospace>Bound A</monospace> holds under the hypothesis that the local repair groups are disjoint and of equal size. Comparing with previous bounds obtained under the same hypothesis, <monospace>Bound A</monospace> either covers them as special cases or has an advantage due to its explicit form. Then, the hypothesis is removed in <monospace>Bound B</monospace> and <monospace>Bound C</monospace>. As the price for explicit form, <monospace>Bound B</monospace> specially holds for <inline-formula> <tex-math notation="LaTeX">$ {d}\geq \text {5}$ </tex-math></inline-formula> and <monospace>Bound C</monospace> for <inline-formula> <tex-math notation="LaTeX">$ {r}=\textbf {2}$ </tex-math></inline-formula>. Through specific comparisons, we show that <monospace>Bound B</monospace> and <monospace>Bound C</monospace> both tend to outperform the C-M bound, as <inline-formula> <tex-math notation="LaTeX">$ {n}$ </tex-math></inline-formula> goes large. Moreover, a family of binary linear LRCs with <inline-formula> <tex-math notation="LaTeX">$ {d}\geq \textbf {6}$ </tex-math></inline-formula> attaining <monospace>Bound B</monospace> are constructed and later extended to a wider range of parameters by a shortening technique. Lastly, most of the bounds and constructions are extended to <inline-formula> <tex-math notation="LaTeX">$ {q}$ </tex-math></inline-formula>-ary LRCs.