Additive codes that beat linear codes
Additive codes over F_{q^h} reach the Griesmer bound for large minimum distance. Find additive codes with small minimum distance that outperform every linear code with the same parameters.
Cite
@misc{cairn-additive-codes-beating-linear,
title = {Additive codes that beat linear codes},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/additive-codes-beating-linear}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
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The problem
The question
An additive code over the alphabet F_{q^h} is closed under addition and under multiplication by the subfield F_q; it has q^k codewords (k may be fractional in units of h). Such codes correspond to multisets of n subspaces of dimension at most h in the projective geometry PG(k − 1, q). Kurz proved that for given q, k, h and large minimum distance d, additive codes attain their Griesmer bound. Open problems from the abstract:
- Find constructions of additive codes that outperform linear codes for relatively small minimum distances.
- Find improved upper bounds (and decrease the parameter needed in Solomon–Stiffler-type constructions).
- Find Griesmer-type bounds and matching constructions for other metrics, e.g. b-symbol codes.
What counts as progress
- A new code: an additive code over, e.g., F_4 or F_9 whose length, size and minimum distance beat the best linear code with the same parameters (compare with known tables).
- New upper bounds with proofs.
How it is checked
A code is a finite object (a generator matrix over F_q, or the list of subspaces); its size and minimum distance are computed exactly by a program (level A).
Source. Posed by Sascha Kurz in an extended abstract of the Oberwolfach workshop New Mathematical Directions in Coding Theory (2025), recorded in Oberwolfach Reports 41/2025, p. 2209 (EMS Press, DOI 10.4171/OWR/2025/41), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.