Algorithm 1059: LinCodeWeightInv—Library for Computing the Weight Distribution of Linear Codes over Finite Fields

Abstract

This paper introduces Algorithm 1059, implemented as the LinCodeWeightInv library, which provides a highly efficient method for computing the weight distribution of linear codes over finite fields. This functionality is crucial for accurately assessing the error-correcting capabilities and determining the minimum distance of modern coding schemes. The library offers researchers and engineers a robust and optimized tool to analyze code performance critical to communication reliability and post-quantum security.

Report

Key Highlights

• Introduction of the new software library, LinCodeWeightInv, embodying Algorithm 1059.
• The primary function is the efficient and precise computation of the weight distribution for linear codes defined over finite fields ($GF(q)$).
• Addresses a fundamental and computationally complex challenge in coding theory, providing a robust, verified algorithmic solution published in ACM Transactions on Mathematical Software.
• The weight distribution data is essential for determining the error correction capabilities (minimum distance) and cryptographic strength of code-based systems.

Technical Details

• Algorithm (Inferred): Algorithm 1059 likely employs highly optimized mathematical techniques, potentially utilizing the MacWilliams Identity or sophisticated search methods (e.g., dynamic programming or iterative pruning) to manage the exponential complexity inherent in weight distribution calculation.
• Scope: Focuses specifically on linear codes, which are foundational in modern communications (e.g., LDPC, Polar Codes) and cryptography (e.g., McEliece schemes).
• Implementation: As a library, LinCodeWeightInv is expected to provide modular, reusable functions optimized for performance, potentially leveraging parallel processing or specialized data structures tailored for finite field arithmetic.

Implications

• PQC Security Analysis: Efficient computation of weight distribution is critical for analyzing the security parameters (specifically the minimum distance) of code-based Post-Quantum Cryptography (PQC) systems, helping validate or break proposed schemes much faster.
• Communication Reliability: Faster analysis of weight distribution allows engineers to more quickly iterate and design highly efficient Error Correction Codes (ECC) for demanding applications, such as deep-space communication, high-speed networking, and reliable data storage.
• RISC-V Acceleration Opportunities: The computational intensity of Algorithm 1059 makes it a prime target for hardware acceleration. Developers can use the library's performance bottlenecks to define custom instructions or specialized accelerator blocks (Co-processors) within the extensible RISC-V ISA, optimizing complex finite field operations and matrix calculations inherent in coding theory.