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Linear Codes with Two or Three Weights From Quadratic Bent Functions
Zhengchun Zhou, Nian Li, Cuiling Fan, Tor Helleseth
TL;DR
Few-weight linear codes matter for several combinatorial and communication-related applications, but constructing such codes with controlled distributions is the paper's focus. The paper uses quadratic Bent functions over Fp to construct two- or three-weight codes and determine their weight distributions. The resulting codes extend earlier constructions, and some meet linear-code bounds, while non-full-rank alternatives may have poor minimum distance.
Problem
Few-weight linear codes have important applications, motivating constructions with controlled weights and weight distributions.
Method
The paper constructs p-ary linear codes from quadratic Bent functions over Fp, using their full-rank structure and associated quadratic-form results.
Results
The constructions produce three-weight codes for odd m and two-weight codes for even m, with weight distributions determined; some codes meet linear-code bounds.
Takeaways & Limitations
The work extends earlier constructions and includes some earlier linear codes as special cases.
Takeaways & Limitations
If the employed quadratic function is not full rank, the corresponding codes may have poor minimum distance; the sign of ε remains open in one construction.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper motivates few-weight linear codes through their applications and introduces a quadratic-Bent-function construction that determines weight distributions and can yield optimal codes.
- Weight distributions determine minimum distance and therefore error-correcting capability, while also supporting error-probability calculations.
- Few-weight linear codes support applications in secret sharing, authentication codes, association schemes, and strongly regular graphs.
- The paper constructs two- or three-weight p-ary linear codes from any quadratic Bent function over Fp.
- The construction includes an earlier construction as a special case and determines the resulting codes' weight distributions.
- Some constructed codes are optimal because they meet bounds for linear codes.
2 Quadratic forms over finite fields
This section develops quadratic forms over finite fields, their ranks and standard types, and the Bent-function characterization used later to construct few-weight codes.
- A quadratic form over Fp is represented as a homogeneous degree-two polynomial after viewing Fpm as an m-dimensional Fp-vector space.
- The rank of a quadratic form is the smallest number of variables needed after nonsingular coordinate transformations, and forms reduce to three standard types.
- Quadratic forms are classified into standard types including Type I, with additional types determined by rank parity and a fixed nonsquare in Fp.
- A Bent function has Walsh-transform magnitude pm/2 at every point, and quadratic forms are Bent exactly when they have full rank.
- The section derives distributions for values associated with quadratic Bent functions by counting solutions and applying quadratic-character formulas.
3 Linear Codes with Two or Three Weights From Quadratic Bent Functions
This section constructs p-ary linear codes from quadratic Bent functions and determines their weight distributions, yielding two-weight codes for even m and three-weight codes for odd m. The construction covers known families, provides trace representations, and includes optimal examples.
- Construction: The paper defines codes from subsets associated with quadratic Bent functions over F_p and determines their weight distributions.The construction follows Ding et al.'s subset-and-trace framework and targets cases where minimum distance and weight distribution can be explicitly settled.
- Main results: Three-weight [p^m−1−1,m] codes arise when m is odd.Theorem 1 states this form and associates its distribution with Table 1.
- Main results: Two-weight [p^m−1+ε(p−1)p^(m−2)−1,m] codes arise when m is even, with ε determined by the quadratic form type.For even m, ε=1 for Type I and ε=−1 for Type III.
- Main results: Any quadratic Bent function over F_p naturally yields a two-weight or three-weight linear code according to the parity of m.The resulting distributions are obtained through quadratic-form lemmas and determine the code dimension.
- Known families: Planar functions produce quadratic Bent component functions, supplying several known code families as special cases.The listed planar functions include x^2 and x^(p^k+1), while the Gold class contains Sidelnikov, Kumar–Moreno, and Kasami cases.
- Examples and optimality: The general results include optimal examples, including [40,5,24],, and codes meeting stated bounds.The examples agree with the predicted weight enumerators and are optimal under the cited ternary-code or Griesmer bounds.
4 Concluding Remarks
Quadratic Bent functions yield linear codes with two or three nonzero weights, with weight distributions determined and some codes meeting bounds. The construction’s distance may be weaker for quadratic functions that are not Bent.
- Quadratic Bent functions construct linear codes with two or three nonzero weights, depending on whether the number of variables is even or odd.
- The weight distributions of the constructed codes are determined.
- Some constructed codes are optimal because their parameters meet certain bounds on linear codes.
- Using a non-full-rank quadratic function, such as a semi-bent function, may result in a corresponding code with poor minimum distance.