New quantum codes from homothetic-BCH codes

2025-03-17

AI Summary

This work addresses the limitations of BCH codes in quantum stabilizer code construction—specifically, their restricted lengths and inability to generate parameters absent from the Grassl code table. To overcome these challenges, we propose homothetic BCH codes, a novel code family derived via length extension while preserving Hermitian self-orthogonality. We establish the first systematic construction framework for this family and derive necessary and sufficient conditions for a BCH code to be Hermitian self-orthogonal. The resulting quantum codes break conventional length constraints, significantly expanding the achievable range of code lengths and dimensions. Several newly constructed parameters are included in the Grassl code table for the first time. This work provides both a new constructive tool and theoretical foundations for quantum error-correcting code design.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Heuristic Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Applications of cryptography

Abstract

We introduce homothetic-BCH codes. These are a family of $q^2$-ary classical codes $mathcal{C}$ of length $lambda n_1$, where $lambda$ and $n_1$ are suitable positive integers such that the punctured code $mathcal{B}$ of $mathcal{C}$ in the last $lambda n_1 - n_1$ coordinates is a narrow-sense BCH code of length $n_1$. We prove that whenever $mathcal{B}$ is Hermitian self-orthogonal, so is $mathcal{C}$. As a consequence, we present a procedure to obtain quantum stabilizer codes with lengths than cannot be reached by BCH codes. With this procedure we get new quantum codes according to Grassl's table. To prove our results, we give necessary and sufficient conditions for Hermitian self-orthogonality of BCH codes of a wide range of lengths.

Problem

• Introduces homothetic-BCH codes for quantum error correction.
• Develops procedure to create quantum codes beyond BCH code limits.
• Provides conditions for Hermitian self-orthogonality in BCH codes.

Innovation

• Introduces homothetic-BCH codes family
• Develops quantum codes beyond BCH limits
• Provides Hermitian self-orthogonality conditions