Shift-XOR Convertible Locally Repairable Codes

Shift-XOR codes employ shift and bitwise exclusiveor (XOR) operations and have been applied in distributed storage systems (DSS) to reduce the encoding/decoding computation cost. In this paper, we study shift-XOR Locally Repairable Codes (LRCs) to reduce the computation costs of encoding, decoding, and repairing. By extending an existing bound for LRCs, we obtain a bound on fault tolerance capability relating to the storage overhead due to shifting. We then provide an explicit construction of shift-XOR LRCs that achieves this bound asymptotically in some cases. The proposed construction has a storage overhead of $O\left(k(n-k)^{2}\right)$ bits, which becomes negligible as sequence length increases. Furthermore, we develop an efficient code conversion framework in the merge regime by leveraging locality and shiftXOR operations. Our conversion method reduces access cost while maintaining low computational complexity.