[Paper Review] Groebner bases and combinatorics for binary codes

M. Borges-Quintana, M. A. Borges-Trenard+2 more|ArXiv.org|Sep 8, 2005

Polynomial and algebraic computation7 references4 citations

TL;DR

This paper introduces a binomial ideal derived from a binary linear code and demonstrates how its Gröbner basis under total degree ordering enables efficient decoding and combinatorial analysis of minimal cycles in associated graphs. The key contribution is an adapted FGLM algorithm that computes the Gröbner basis in increasing order of term degree, allowing systematic discovery of minimal codewords and cycle bases with computational advantages for code and graph properties.

ABSTRACT

In this paper we introduce a binomial ideal derived from a binary linear code. We present some applications of a Gröbner basis of this ideal with respect to a total degree ordering. In the first application we give a decoding method for the code. By associating the code with the set of cycles in a graph, we can solve the problem of finding all codewords of minimal length (minimal cycles in a graph), and show how to find a minimal cycle basis. Finally we discuss some results on the computation of the Gröbner basis.

Motivation & Objective

• To establish a connection between binary linear codes and polynomial ideals using binomial generators.
• To develop a decoding algorithm based on Gröbner basis reduction to canonical form.
• To apply Gröbner basis techniques to find all minimal cycles and a minimal cycle basis in graphs associated with codes.
• To design an efficient algorithm for computing the Gröbner basis of the code ideal with inherent ordering advantages.

Proposed method

• Define a monoid over variables $x_1, \dots, x_n$ and map it to $\mathbb{F}_2^n$ via a morphism $\psi$, identifying words with vectors.
• Construct a binomial ideal $I(\mathcal{C})$ generated by $\{w_i - 1\}$ for codewords $w_i$ and $\{x_i^2 - 1\}$ to enforce binary structure.
• Use a total degree-compatible monomial ordering (e.g., degrevlex) to compute a Gröbner basis of $I(\mathcal{C})$, which encodes code properties.
• Apply an adapted FGLM-style algorithm that processes terms in increasing degree order, maintaining a list of representatives and canonical forms.
• Leverage the ordering to detect minimal-weight codewords and cycles early in the computation, enabling efficient extraction of minimal structures.
• Use the Gröbner basis for decoding by reducing received words to their unique normal form via reduction modulo the basis.

Experimental results

Research questions

• RQ1 How can Gröbner bases of binomial ideals be used to decode binary linear codes efficiently?
• RQ2 What combinatorial structures in a graph (e.g., minimal cycles) can be extracted from the Gröbner basis of the code ideal?
• RQ3 Can the FGLM algorithm be adapted to compute Gröbner bases for code ideals in a way that naturally orders results by term degree?
• RQ4 How does the structure of the Gröbner basis reflect the minimal distance and cycle basis of the code or associated graph?
• RQ5 What computational advantages arise from using degree-compatible orderings and early termination in the basis computation?

Key findings

• The Gröbner basis of the ideal $I(\mathcal{C})$ under degrevlex ordering enables decoding via canonical form reduction, with unique solutions for errors of weight $\leq t$.
• The algorithm computes the Gröbner basis in increasing order of term degree, allowing early detection of minimal-weight codewords and minimal cycles.
• The adapted FGLM algorithm efficiently computes the Gröbner basis by maintaining a list of canonical forms and inserting successors in degree order, with pruning via membership checks.
• The method identifies all minimal cycles in the graph associated with the code by extracting basis elements of minimal total degree.
• The reduced Gröbner basis $\{x_2 - x_1, x_3 - x_1, x_1^2 - 1\}$ was computed for a specific code, demonstrating the algorithm's correctness and efficiency.
• The approach allows computation of a minimal cycle basis by selecting cycles of minimal length in the basis, leveraging the degree-ordered traversal.