IEEE Project Abstract

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