Linear Codes from Some 2-Designs (1503.06511v1)

Abstract: A classical method of constructing a linear code over $\gf(q)$ with a $t$-design is to use the incidence matrix of the $t$-design as a generator matrix over $\gf(q)$ of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of $2$-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of $2$-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented.

• The paper presents a novel linear code construction method leveraging specific 2-design techniques including almost difference sets and semibent functions.
• It demonstrates the derivation of one-, two-, and three-weight codes with optimal parameters for applications in secret sharing and secure communications.
• The work bridges combinatorial design theory and coding by offering fresh insights into building codes with enhanced performance and security.

Overview of "Linear Codes from Some 2-Designs" by Cunsheng Ding

This paper presents a novel methodology for constructing linear codes using specific classes of 2-designs, diverging from the classical approach of utilizing incidence matrices. The paper focuses on deriving linear codes with few weights from constructs such as almost difference sets, difference sets, and 2-designs linked to semibent functions. Two families of codes discovered through this method exhibit optimal characteristics.

The framework leads to one-weight, two-weight, and three-weight codes with broad applications in secret sharing, authentication schemes, consumer electronics, communication, and data storage. These linear codes derived from 2-design perspectives also contribute to the characterization of highly nonlinear Boolean functions within coding theory.

Key Mathematical Foundations:

• Almost Difference Sets and Difference Sets: The paper defines almost difference sets and difference sets in the context of abelian groups. It introduces specific conditions under which these structures can be used to generate linear codes. A particular focus is placed on cyclic difference sets, which automatically define linear codes.
• Group Characters in GF(q): The role of additive characters in constructing codes from the defining set $D \subseteq \text{GF}(q)$ is examined, leveraging the canonical additive character and trace functions.

Generic Construction of Linear Codes:

The paper proposes a construction method where a linear code CD is defined from a set $D \subseteq \text{GF}(q)$. The dimension of these codes is generally at most m, with weights derivable through examined properties of trace functions associated with elements in $D$.

Linear Codes from Specific Designs:

• Skew Sets: The paper provides a theorem that establishes the parameters for one-weight codes derived from skew sets—a special type of difference set.
• Images of Functions: Codes generated from the images of functions $f : \text{GF}(q) \to \text{GF}(q)$ are discussed, focusing on the case of quadratic functions to determine weight distributions under defined conditions.
• Preimages of Functions: The paper also considers codes based on the supports (preimages) of certain functions, revealing structures with potentially useful coding parameters.

Implications and Future Directions:

The results from Cunsheng Ding’s research suggest several practical implications. The multi-weight linear codes enrich the toolbox for designing robust authentication codes and enable more efficient data transmission mechanisms, potentially impacting IT security strategies and digital communication protocols.

Theoretically, the work adds to the understanding of nonlinear Boolean functions' algebraic properties, offering pathways for future paper into optimizing linear code parameters and uncovering novel applications. Further exploration into more complex classes of designs could unravel additional coding schemes with advantageous characteristics suitable for evolving technological demands.

In conclusion, the paper enriches the field of coding theory by blending the theory of combinatorial designs with modern coding challenges, fostering advancements potential in both secure communications and nonlinear Boolean function characterization.