[Paper Review] Further Results on Homogeneous Two-Weight Codes

Thomas Honold|arXiv (Cornell University)|Jan 28, 2014

Coding theory and cryptography8 references3 citations

TL;DR

This paper extends the correspondence between linear two-weight codes and strongly regular graphs to modular two-weight codes over finite Frobenius rings. It proves that such codes induce strongly regular Cayley graphs via coset structures, and constructs a dual code with analogous two-weight and strongly regular graph properties, generalizing classical results from finite fields to a broader algebraic setting.

ABSTRACT

The results of [1,2] on linear homogeneous two-weight codes over finite Frobenius rings are exended in two ways: It is shown that certain non-projective two-weight codes give rise to strongly regular graphs in the way described in [1,2]. Secondly, these codes are used to define a dual two-weight code and strongly regular graph similar to the classical case of projective linear two-weight codes over finite fields [3].

Motivation & Objective

• To generalize the known correspondence between projective linear two-weight codes over finite fields and strongly regular graphs to a broader class of codes over finite Frobenius rings.
• To establish that non-projective, modular two-weight codes over Frobenius rings give rise to strongly regular Cayley graphs through their coset structures.
• To define a dual code for modular two-weight codes and prove that this dual also forms a two-weight code with an associated strongly regular graph.
• To unify and generalize classical results on two-weight codes and partial difference sets in the context of Frobenius rings.

Proposed method

• Uses the homogeneous weight function on finite Frobenius rings, defined via characters and invariant under unit multiplication.
• Introduces modular two-weight codes as a generalization of projective two-weight codes, characterized by specific weight distributions and non-degeneracy conditions.
• Defines the Cayley graph Γ(C) over the quotient group C/C₀, where vertices are cosets and adjacency is determined by homogeneous weight w₁.
• Applies the correlation property of homogeneous weights (Equation 4) to compute the number of common neighbors in the Cayley graph.
• Constructs the dual code C′ as the right linear code generated by the columns of a matrix formed from codewords of weight w₁.
• Uses the structure of the column space D of the generator matrix to relate the dual code to partial difference sets in (D, +).

Experimental results

Research questions

• RQ1 Can the correspondence between two-weight codes and strongly regular graphs be extended beyond projective codes over finite fields to modular two-weight codes over Frobenius rings?
• RQ2 Under what conditions does a modular two-weight code over a Frobenius ring give rise to a strongly regular Cayley graph?
• RQ3 Does the dual code of a modular two-weight code inherit the two-weight and strongly regular graph properties?
• RQ4 What is the precise parameterization of the strongly regular graph associated with a modular two-weight code and its dual?
• RQ5 How do the properties of the homogeneous weight and the structure of the ring influence the existence and parameters of such graphs?

Key findings

• The Cayley graph Γ(C) associated with a modular two-weight code C over a finite Frobenius ring is strongly regular with parameters explicitly computed in terms of |C|, w₁, w₂, and n.
• The graph Γ(C) is trivial if and only if w₁ = n, corresponding to the case where codewords of weight 0 and w₂ form a linear subcode.
• The dual code C′ of a modular two-weight code C with C₀ = {0} is also a modular two-weight code with weights w₁′ = b₁w₁/n and w₂′ = (w₂ − n)|C|/(w₂ − w₁).
• The dual graph Γ(C′) is strongly regular with parameters N′ = |C|, K′ = n/r, λ′ = (2n − w₁ − w₂)/r + w₁w₂/(r²|C|), and μ′ = w₁w₂/(r²|C|), where r is the rank of the column space.
• A homogeneous two-weight code C is equivalent to the set Ω = ∪gᵢR× being a regular partial difference set in the column space D of its generator matrix, provided Ω ∪ {0} is not a submodule.
• The condition that Ω ∪ {0} is not a submodule ensures that C is not a one-weight code, and D ∖ Ω is a submodule if and only if w₁ = n, which corresponds to triviality of the graph.