Linear Codes with Two or Three Weights From Quadratic Bent Functions (1506.06830v2)

Abstract: Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, several classes of $p$-ary linear codes with two or three weights are constructed from quadratic Bent functions over the finite field $\gf_p$, where $p$ is an odd prime. They include some earlier linear codes as special cases. The weight distributions of these linear codes are also determined.

• The paper constructs linear codes with two or three weights by utilizing the properties of quadratic Bent functions over finite fields.
• The construction shows that codes derived from quadratic Bent functions exhibit two weights if the dimension m is even and three weights if m is odd.
• The paper derives specific weight distributions for these codes, highlighting their potential applications in areas like secret sharing and authentication codes.

Linear Codes with Two or Three Weights From Quadratic Bent Functions

The paper explores the construction and analysis of linear codes with specific weight distributions derived from quadratic Bent functions over finite fields. The primary focus is on $p$-ary linear codes with two or three nonzero weights, which possess applications in areas such as secret sharing, authentication codes, association schemes, and strongly regular graphs.

Construction of Linear Codes

Central to the paper is the development of linear codes using quadratic Bent functions. A Bent function is characterized by its maximal distance from linear functions, providing optimal nonlinearity. The authors define a linear code $C_D$ of length $n$ from a subset $D$ of the finite field $F_{p^m}$. When the subset $D$ is derived from quadratic Bent functions, they show that such codes exhibit two or three weights, depending upon whether the dimension $m$ is even or odd, respectively. This approach includes previously known linear codes as special cases.

Theoretical Foundations and Results

The theoretical underpinning involves quadratic forms over finite fields and their rank, which influence the weight distribution of the resultant codes. The paper utilizes properties such as equivalence to standard types and number of solutions to specific quadratic equations. The linkage between quadratic forms and Bent functions is rigorously established, providing the foundation for constructing the described codes.

Results on Weight Distribution

The paper meticulously derives the weight distributions for the constructed linear codes in various scenarios:

• Odd $m$:

An [pm − 1, m] three-weight code is produced, with a specific weight distribution determined by the rank of the quadratic form.

• Even $m$:

The constructed linear code becomes a two-weight code, corresponding to different equivalency types of the quadratic form.

The weight distributions, crucial for determining the error-correcting capabilities and the theoretical performance of the codes, are explicitly delineated for both cases.

Applications and Implications

The constructed codes provide optimal or nearly optimal parameters, meeting known bounds on linear codes such as the Griesmer bound, in certain instances. The work promises practical relevance in the design of error-correcting codes for communication systems and secure information exchange protocols.

Future Directions and Open Questions

The methods outlined enable the construction of linear codes using any quadratic function over finite fields. However, codes derived from non-full rank quadratic functions may not exhibit good minimal distances, underlining the importance of Bent functions in the construction process.

The paper leaves an open question regarding the exact determination of the sign of ε for certain quadratic Bent functions, which could further refine the weight distribution results. This invites continued exploration into the equivalence and characterization of these special functions, with ramifications for the wider field of coding theory.

In conclusion, the paper enriches the body of knowledge on linear codes by leveraging the properties of quadratic Bent functions, setting the stage for further exploration into optimal coding strategies and their applications in modern theoretical and practical domains.