Bidirectional Piggybacking Design for Systematic Nodes with Sub-Packetization $l=2$

In 2013, Rashmi et al. proposed the piggybacking design framework to reduce the repair bandwidth of $(n,k;l)$ MDS array codes with small sub-packetization $l$ and it has been studied extensively in recent years. In this work, we propose an explicit bidirectional piggybacking design (BPD) with sub-packetization $l=2$ and the field size $q=O(n^{\lfloor r/2 \rfloor \!+\!1})$ for systematic nodes, where $r=n-k$ equals the redundancy of an $(n,k)$ linear code. And BPD has lower average repair bandwidth than previous piggybacking designs for $l=2$ when $r\geq 3$. Surprisingly, we can prove that the field size $q\leq 256$ is sufficient when $n\leq 15$ and $n-k\leq 4$. For example, we provide the BPD for the $(14,10)$ Reed-Solomon (RS) code over $\mathbb{F}_{2^8}$ and obtain approximately $41\%$ savings in the average repair bandwidth for systematic nodes compared with the trivial repair approach. This is the lowest repair bandwidth achieved so far for $(14,10)_{256}$ RS codes with sub-packetization $l=2$.