A Group Theoretic Construction of Batch Codes
Eldho K. Thomas
TL;DR
The paper addresses the need for batch codes that support arbitrary batch sizes while decoupling design from the code dimension, proposing a group-theoretic construction based on quasi-uniform codes derived from abelian groups, especially $2$-groups. It develops encoding/decoding via group homomorphisms and coset intersections, yielding near-optimal code lengths that scale with the batch size $t$ and not strictly with $k$. Concrete realizations are given for $G=(\mathbb{Z}_2)^3$ and $G=(\mathbb{Z}_2)^4$, including a $(65,4,32)$ batch code, and generalizations to $G=(\mathbb{Z}_2)^k$ with distinct treatments for odd and even $k$ based on subspace combinatorics and complementarity. The framework also highlights potential dual use as locally repairable codes and PIR-like schemes, offering a versatile paradigm for designing coded storage systems with algebraic structure and practical decoding simplicity.
Abstract
Batch codes serve as critical tools for load balancing in distributed storage systems. While numerous constructions exist for specific batch sizes t, current methodologies predominantly rely on code dimension parameters, limiting their adaptability. Practical implementations, however, demand versatile batch code designs capable of accommodating arbitrary batch sizes-a challenge that remains understudied in the literature. This paper introduces a novel framework for constructing batch codes through finite groups and their subgroup structures, building on the quasi-uniform group code framework proposed by Chan et al. By leveraging algebraic properties of groups, the proposed method enables systematic code construction, streamlined decoding procedures, and efficient reconstruction of information symbols. Unlike traditional linear codes, quasi-uniform codes exhibit broader applicability due to their inherent structural flexibility. Focusing on abelian 2-groups, the work investigates their subgroup lattices and demonstrates their utility in code design-a contribution of independent theoretical interest. The resulting batch codes achieve near-optimal code lengths and exhibit potential for dual application as locally repairable codes (LRCs), addressing redundancy and fault tolerance in distributed systems. This study not only advances batch code construction but also establishes group-theoretic techniques as a promising paradigm for future research in coded storage systems. By bridging algebraic structures with practical coding demands, the approach opens new directions for optimizing distributed storage architectures.