Lower Bounds for Error Coefficients of Griesmer Optimal Linear Codes via Iteration

2025-07-07

AI Summary

Determining tight lower bounds on the error coefficient—i.e., the number of minimum-weight codewords—of Griesmer-optimal linear codes remains challenging, especially when linear programming bounds become ineffective. To address this, we propose a novel lower-bounding method based on iterative analysis and combinatorial construction. Our approach integrates minimum-weight codeword counting theory with the Griesmer bound, thereby overcoming performance limitations inherent in conventional linear programming bounds. For code dimensions up to five, the derived bounds are tight in most cases; where not tight, the deviation from the true error coefficient is at most two, indicating near-optimality. These results significantly enhance the predictability and theoretical accuracy of asymptotic frame error rates for linear codes over additive white Gaussian noise (AWGN) channels.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Information Theory

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The error coefficient of a linear code is defined as the number of minimum-weight codewords. In an additive white Gaussian noise channel, optimal linear codes with the smallest error coefficients achieve the best possible asymptotic frame error rate (AFER) among all optimal linear codes under maximum likelihood decoding. Such codes are referred to as AFER-optimal linear codes. The Griesmer bound is essential for determining the optimality of linear codes. However, establishing tight lower bounds on the error coefficients of Griesmer optimal linear codes is challenging, and the linear programming bound often performs inadequately. In this paper, we propose several iterative lower bounds for the error coefficients of Griesmer optimal linear codes. Specifically, for binary linear codes, our bounds are tight in most cases when the dimension does not exceed $5$. To evaluate the performance of our bounds when they are not tight, we also determine the parameters of the remaining 5-dimensional AFER-optimal linear codes. Our final comparison demonstrates that even when our bounds are not tight, they remain very close to the actual values, with a gap of less than or equal to $2$.

Problem

• Establish tight lower bounds for Griesmer optimal linear codes' error coefficients
• Improve upon inadequate linear programming bounds via iterative methods
• Evaluate bounds' accuracy for 5-dimensional AFER-optimal binary codes

Innovation

• Iterative lower bounds for error coefficients
• Tight bounds for binary codes up to dimension 5
• Close approximation with gaps ≤ 2