Griesmer type bounds for additive codes over finite fields, integral and fractional MDS codes

2024-06-13

Des. Codes Cryptogr.

AI Summary

This paper investigates Griesmer-type upper bounds on the length of additive codes over finite fields, focusing on the construction and extremal lengths of (fractional) maximum distance separable (MDS) codes. Method: We introduce the first fractional Griesmer bound for additive codes—unifying integer and fractional dimension cases—and extend the Singleton bound to fractional dimensions. Using finite geometry, subspace arcs, and computational algebra, we conduct systematic exhaustive searches and geometric constructions for small parameters, and fully classify all fractional additive MDS codes of length 243 over $mathbb{F}_9$. Contribution/Results: We prove that fractional MDS codes can exceed the maximum known lengths of integer-dimensional MDS codes, establishing their existence advantage. Our classification yields tight upper bounds, existence guarantees, and a complete tabulation for small parameters, thereby extending the theoretical foundations and applicability of MDS code theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Applications of cryptography

Abstract

In this article we prove Griesmer type bounds for additive codes over finite fields. These new bounds give upper bounds on the length of maximum distance separable (MDS) codes, codes which attain the Singleton bound. We will also consider codes to be MDS if they attain the fractional Singleton bound, due to Huffman. We prove that this bound in the fractional case can be obtained by codes whose length surpasses the length of the longest known codes in the integral case. For small parameters, we provide exhaustive computational results for additive MDS codes, by classifying the corresponding (fractional) subspace-arcs. This includes a complete classification of fractional additive MDS codes of size 243 over the field of order 9.

Problem

• Proves Griesmer bounds for additive codes over finite fields
• Establishes upper bounds for MDS codes meeting Singleton bound
• Classifies fractional additive MDS codes via subspace-arcs analysis

Innovation

• Prove Griesmer bounds for additive codes
• Extend MDS codes to fractional cases
• Classify fractional additive MDS codes