Construction of locally repairable codes based on iterative matrix

Abstract:

To address the issues of insufficient parameter flexibility and low code rate in locally repairable codes (LRCs) within current distributed storage systems, this paper introduces two new types of locally repairable codes. First, this paper constructs a class of iterative matrices based on the combination of all-zero matrices and all-one vectors, and then proposes a construction algorithm for all symbol-locally repairable codes (AS-LRCs) with (r,t)-locality using the constructed iterative matrices as parity-check matrices. Subsequently, by improving the structure of the parity-check matrix of AS-LRCs based on iterative matrices, a construction algorithm for information symbol-locally repairable codes (IS-LRCs) with (r,t)-locality is further proposed. Experimental and theoretical analyses show that AS-LRCs meet strict availability requirements, and when the availability parameter t=2, their code length reaches the theoretical minimum bound, making them the optimal LRCs in terms of code length. The minimum distance of IS-LRCs reaches the Singleton-like optimal bound, making them the optimal LRCs in terms of minimum distance. Both AS-LRCs and IS-LRCs construction algorithms support flexible configuration of arbitrary locality and availability. The code rates of the two construction algorithms are significantly higher than existing methods, reaching the theoretical optimal bound of code rate when t=2. The construction algorithms of the two types of LRCs not only ensure efficient data repair but also support more flexible parameter configurations and achieve higher code rates. This provides more efficient coding strategies for distributed storage systems and thereby enhancing the overall performance of distributed storage systems.