Linear Codes Constructed from Two Weakly Regular Plateaued Functions with Index (p - 1)/2

Area of Science:

• Coding Theory
• Cryptography
• Number Theory

Background:

• Linear codes are fundamental in cryptography and coding theory.
• Codes with few weights have diverse applications, including authentication and secret sharing.
• Plateaued functions are crucial in constructing cryptographic primitives.

Purpose of the Study:

• To construct an infinite family of linear codes with few weights.
• To analyze the properties of these codes, specifically their weight distributions.
• To explore the applicability of these codes in cryptographic and theoretical contexts.

Main Methods:

• Construction of linear codes using two distinct weakly regular unbalanced (and balanced) plateaued functions.
• Setting the parameter p such that p ≡ 1 (mod 4).
• Utilizing exponential sums and Walsh transform for analysis.

Main Results:

• An infinite family of linear codes was successfully constructed.
• The weight distributions of the constructed codes were completely determined.
• Most of the codes exhibit few nonzero weights and are minimal.

Conclusions:

• The study presents a novel construction for linear codes with desirable properties.
• The findings contribute to the understanding of code properties and their applications.
• The constructed codes offer potential for advancements in cryptography and coding theory.