Advances in Quantum Error Correction and Coding Theory

The field of quantum error correction and coding theory is rapidly advancing, with a focus on developing new codes and techniques that can efficiently correct errors in quantum computations. Recent research has explored the use of generalized bicycle codes, which have shown promising results in terms of error threshold performance and encoding efficiency. Additionally, there has been significant progress in the construction of batch codes, which are critical for load balancing in distributed storage systems. Group-theoretic techniques have emerged as a powerful tool for constructing and analyzing codes, and have been successfully applied to various coding problems. Notable papers in this area include those that introduce new families of codes with improved parameters, such as the generalized Kitaev codes, and those that develop novel frameworks for constructing batch codes. For example, one paper introduces a family of codes that generalizes both standard and optimized Kitaev codes, with parameters of the form [| 2n, 2, ≥ √n |] where n is a factor of 1 + d^2. Another paper presents a group-theoretic construction of batch codes, which enables systematic code construction, streamlined decoding procedures, and efficient reconstruction of information symbols. These developments have the potential to significantly impact the design of quantum error correction codes and distributed storage systems.

Sources

On the Generalization of Kitaev Codes as Generalized Bicycle Codes

A Group Theoretic Construction of Batch Codes

On Weight Enumeration and Structure Characterization of Polar Codes via Group Actions

On the Complexity of Identifying Groups without Abelian Normal Subgroups: Parallel, First Order, and GI-Hardness

Skew generalized quasi-cyclic codes over non-chain ring $F_q+vF_q$

Computing change of level and isogenies between abelian varieties

Enumeration of minimum weight codewords of affine Cartesian codes

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