Construction of Binary Locally Repairable Codes with Nonuniform Locality and Availability Using Combinatorial...

Area of Science:

• Coding Theory
• Combinatorial Design Theory

Background:

• Locally Repairable Codes (LRCs) are crucial for distributed storage systems, enabling efficient data recovery.
• Existing constructions often lack flexibility in locality or availability, and binary implementations are preferred for practical use.

Purpose of the Study:

• To generalize the construction of LRCs using Pairwise Balanced Designs (PBDs) and Balanced Incomplete Block Designs (BIBDs).
• To develop binary LRCs with nonuniform locality or availability, achieving optimal or near-optimal performance.
• To address open problems, such as constructions where r does not divide k.

Main Methods:

• Leveraging PBDs and BIBDs for code construction.
• Prioritizing binary code implementations.
• Analyzing code rate, minimum distance, and repair efficiency.

Main Results:

• Proposed distance-optimal LRCs with nonuniform localities and message-symbol (r,t)-availability.
• Achieved optimal minimum distance under known bounds for binary codes.
• Demonstrated higher code rates compared to existing constructions.
• Showcased constructions where r does not divide k.

Conclusions:

• The proposed PBD and BIBD-based constructions provide a powerful framework for generalized LRCs.
• These binary codes offer superior performance and broader applicability, outperforming prior works.
• The study advances the field of coding theory with practical and efficient code designs.