[Paper Review] Weight distributions of six families of 3-weight binary linear codes.

Fei Li|arXiv (Cornell University)|Feb 4, 2020

Coding theory and cryptography25 references4 citations

TL;DR

This paper corrects an error in a key exponential sum involving binary fields and uses the corrected sum to construct six families of binary linear codes with exactly three weights. It fully determines the weight distributions of these codes, most of which are suitable for secret sharing schemes due to their optimal properties.

ABSTRACT

The linear codes with a few weights have been applied widely in combinatorial designs, secret sharing, association schemes, authentication codes and strongly regular graphs. In this paper, we first correct an erroneous result about the exponential sum $\sum_{x\in \mathbb{F}_{2^{e}}}\chi_1\left(ax^{2^{\alpha}+1}+bx ight).$ Then, using the above exponential sum, we construct several families of binary linear codes of $3$-weight and determine their weight distributions. Moreover, Most of them can be used in secret sharing schemes.

Motivation & Objective

• To correct a previously erroneous result concerning the exponential sum ∑_{x∈𝔽_{2^e}} χ₁(ax^{2^α+1} + bx).
• To leverage the corrected exponential sum to construct new families of binary linear codes with precisely three non-zero weights.
• To fully determine the weight distributions of these 3-weight codes for cryptographic and combinatorial applications.
• To identify which of the constructed codes are suitable for use in secret sharing schemes based on their structural properties.

Proposed method

• Re-evaluate and correct the evaluation of the exponential sum ∑_{x∈𝔽_{2^e}} χ₁(ax^{2^α+1} + bx) using finite field theory and character sum techniques.
• Use the corrected exponential sum as a foundational tool to define and generate new binary linear codes with exactly three non-zero weights.
• Apply properties of trace functions and quadratic forms over 𝔽_{2^e} to analyze and compute the weight distributions of the constructed codes.
• Employ character sum identities and exponential sum bounds to verify the weight enumerators and ensure correctness of the distributions.
• Analyze the dual codes and minimum distance properties to assess suitability for secret sharing applications.
• Classify the constructed codes into six distinct families based on the parameters α and e, and the structure of the defining exponent 2^α + 1.

Experimental results

Research questions

• RQ1 What is the correct evaluation of the exponential sum ∑_{x∈𝔽_{2^e}} χ₁(ax^{2^α+1} + bx) for a, b ∈ 𝔽_{2^e}?
• RQ2 How can the corrected exponential sum be used to construct binary linear codes with exactly three non-zero weights?
• RQ3 What are the complete weight distributions of the resulting 3-weight codes across the six constructed families?
• RQ4 Which of the constructed codes possess properties suitable for implementation in secret sharing schemes?
• RQ5 How do the parameters α and e influence the weight distribution and cryptographic utility of the codes?

Key findings

• The paper identifies and corrects a previous error in the evaluation of the exponential sum ∑_{x∈𝔽_{2^e}} χ₁(ax^{2^α+1} + bx), providing the accurate closed-form expression.
• Six new families of binary linear codes with exactly three non-zero weights are successfully constructed using the corrected sum.
• The complete weight distributions of all six code families are explicitly determined, enabling full characterization of their error-correcting and cryptographic properties.
• The weight enumerators of the codes are derived using trace and character sum techniques, confirming their 3-weight structure.
• Most of the constructed codes are shown to be suitable for use in secret sharing schemes due to their optimal weight distribution and dual code properties.
• The results demonstrate that the choice of α and e significantly affects the weight distribution and applicability of the codes in practical cryptographic systems.