Optimal Locally Repairable Codes With Multiple Erasure Tolerance

To address the issues of stringent parameter constraints and lower code rates in the construction of locally repairable codes (LRCs) under multiple erasures, this letter proposes (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i>, δ, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>)-LRCs with flexible availability and hierarchical LRCs (H-LRCs) with flexible levels, based on resolvable group divisible design (RGDD). Specifically, using cyclic planar difference sets to obtain the triple of RGDD, LRCs with (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i>, δ, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>)-information-locality are constructed based on the parallel class blocks of RGDD. Furthermore, H-LRCs are designed for information symbols with parameters ( (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>, δ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub>), (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i><sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub>, δ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub>), . . . , (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r_h</i>, δ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>h</i></sub>) ) by combining the elements within RGDD groups. Both the constructed LRCs and H-LRCs achieve optimal minimum distances. Compared to existing (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i>, δ, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>)-LRCs, the proposed (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i>, δ, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>)-LRCs provide more flexible availability, fewer parameter constraints, and higher code rates. In contrast to current H-LRCs, the proposed H-LRCs impose fewer parameter constraints and allow the number of levels to be adjusted according to the number of information nodes. Moreover, the proposed H-LRCs optimize the parameters of (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i>, δ, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</i>)-LRCs, resulting in higher code rates.