[Paper Review] A new upper bound for subspace codes

Daniel Heinlein, Sascha Kurz|arXiv (Cornell University)|Mar 25, 2017

Cooperative Communication and Network Coding13 references3 citations

TL;DR

This paper establishes a new upper bound of 272 for the maximum size of a binary constant dimension subspace code with packet length 8, minimum subspace distance 6, and dimension 4, using integer linear programming (ILP) combined with classification results of smaller subspace codes. The result improves upon the previous bound of 289 by leveraging symmetry reduction through substructure enumeration and ILP formulation under structural constraints.

ABSTRACT

It is shown that the maximum size $A_2(8,6;4)$ of a binary subspace code of packet length $v=8$, minimum subspace distance $d=4$, and constant dimension $k=4$ is at most $272$. In Finite Geometry terms, the maximum number of solids in $\operatorname{PG}(7,2)$, mutually intersecting in at most a point, is at most $272$. Previously, the best known upper bound $A_2(8,6;4)\le 289$ was implied by the Johnson bound and the maximum size $A_2(7,6;3)=17$ of partial plane spreads in $\operatorname{PG}(6,2)$. The result was obtained by combining the classification of subspace codes with parameters $(7,17,6;3)_2$ and $(7,34,5;\{3,4\})_2$ with integer linear programming techniques. The classification of $(7,33,5;\{3,4\})_2$ subspace codes is obtained as a byproduct.

Motivation & Objective

• To improve the upper bound on the maximum size of binary constant dimension subspace codes with parameters (8, N, 6; 4)₂.
• To reduce the gap between known lower and upper bounds for A₂(8,6;4) by leveraging structural constraints from smaller codes.
• To develop and apply an ILP-based framework that exploits symmetry by fixing substructures derived from classification results.
• To classify all (7,33,5;{3,4})₂ subspace codes up to isomorphism as a byproduct of the analysis.
• To validate the bound through alternative approaches and provide a foundation for future improvements in constant dimension code bounds.

Proposed method

• Utilized classification results of (7,17,6;3)₂ and (7,16,6;4)₂ subspace codes as fixed substructures in the ILP formulation.
• Applied integer linear programming (ILP) to compute the maximum number of 4-dimensional subspaces in PG(7,2) under subspace distance constraints.
• Prescribed the presence of a (7,17,6;3)₂ code and a (7,16,6;4)₂ code in a hyperplane to break symmetry and improve ILP solvability.
• Defined an ILP formulation with binary variables for 4-dimensional subspaces and constraints on incidences with 1- and 7-dimensional subspaces.
• Used incidence constraints: ∑x_U ≤ 17 for 1- and 7-dimensional subspaces, ∑x_U ≤ 1 for 2- and 6-dimensional subspaces, and a global sum constraint on projective points.
• Employed automorphism group reduction by fixing known subcodes, thereby reducing symmetry and enabling efficient ILP solving.

Experimental results

Research questions

• RQ1 Can the upper bound for A₂(8,6;4) be improved beyond the Johnson bound value of 289?
• RQ2 What is the maximum number of 4-dimensional subspaces in PG(7,2) such that any two intersect in at most a point?
• RQ3 Can integer linear programming techniques be effectively applied to subspace codes when symmetry is high?
• RQ4 What structural subcodes can be used to reduce symmetry and enable feasible ILP computation in large subspace code problems?
• RQ5 Are there (7,33,5;{3,4})₂ subspace codes, and if so, how many are there up to isomorphism?

Key findings

• The maximum size of a binary constant dimension code with parameters (8, N, 6; 4)₂ is at most 272, improving the prior upper bound of 289.
• The bound was achieved by combining classification results of (7,17,6;3)₂ and (7,16,6;4)₂ codes with symmetry-reducing ILP formulations.
• The ILP formulation was solved successfully in 563 cases, each yielding z(F₃, F₄) ≤ 272, confirming the upper bound.
• A complete classification of (7,33,5;{3,4})₂ subspace codes up to isomorphism was obtained as a byproduct.
• The result implies that the maximum number of solids in PG(7,2) mutually intersecting in at most a point is at most 272.
• The approach demonstrates that symmetry reduction via substructure fixing enables effective ILP application to large subspace code problems.