Decoding Desarguesian spread codes beyond half minimum distance

2026-07-18

AI Summary

This work addresses the problem of unique decoding for Desarguesian spread codes under insertions and deletions that alter subspace dimensions, particularly when the total number of errors exceeds half the minimum distance. By integrating coding theory in the subspace metric, geometric structural analysis, and algebraic decoding techniques, the authors establish—for the first time—theoretical guarantees and an effective decoder capable of reliable unique decoding provided the number of deleted dimensions does not exceed \(k-2\). Building on this foundation, two algorithmic enhancements are proposed, which empirically approach the optimal tolerance to insertions. Experimental results demonstrate that the proposed methods maintain a low decoding failure rate even when the total error count surpasses half the minimum distance, as long as the deletion dimension constraint is satisfied, thereby significantly extending the error-correction capability beyond existing decoders.

Abstract

Spread codes are a well-known family of constant-dimension subspace-metric codes. For constant dimension $k$ and ambient space dimension $n$ being a multiple of $k$, these codes have minimum distance $2k$ and a rich geometric structure. In this paper, we study the decoding capabilities of the Nearest Neighbor Decoder for Desarguesian spread codes, establishing that unique decoding is still achievable beyond half the minimum distance. Motivated by this, we develop a new decoding algorithm to uniquely decode Desarguesian spread codes in the presence of both insertions and deletions, which increase and decrease, respectively, the dimension of the transmitted codeword. Even when the sum of the dimensions of insertions and deletions exceeds half the minimum distance, provided that deletions are of dimension at most $k-2$, the algorithm succeeds with a small decoding failure. We also propose two refinements to this algorithm that, empirically, can handle nearly as many insertions as the Nearest Neighbor Decoder.

Problem

• spread codes
• subspace-metric codes
• decoding beyond half minimum distance
• insertions and deletions
• Desarguesian spread codes

Innovation

• Desarguesian spread codes
• subspace-metric codes
• insertion-deletion errors
• unique decoding beyond half minimum distance
• nearest neighbor decoding