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A Class of Three-Weight Cyclic Codes
Zhengchun Zhou, Cunsheng Ding
TL;DR
The paper addresses the construction and characterization of three-weight cyclic codes over GF(p), for which weight distributions and optimality require systematic analysis. It defines a class through minimal polynomials and evaluates the associated sums using quadratic-form methods. The resulting codes have determined weight distributions, some are optimal, and a subclass of their duals is also optimal.
Problem
Three-weight cyclic codes are relatively scarce despite potential applications in association schemes and secret sharing, motivating the construction of additional examples.
Method
The paper constructs cyclic codes over GF(p) using parity-check polynomial h1(x)h2(x) and determines their weights through value distributions of S(a,b) and T(a,b).
Results
The constructed codes have three nonzero weights under the stated parameter conditions, with their weight distributions determined and explicit parameter families obtained.
Takeaways & Limitations
Some of the cyclic codes are optimal, and the duals of a subclass are optimal as well.
Abstract
from arXiv · showhide
Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a class of three-weight cyclic codes over $\gf(p)$ whose duals have two zeros is presented, where $p$ is an odd prime. The weight distribution of this class of cyclic codes is settled. Some of the cyclic codes are optimal. The duals of a subclass of the cyclic codes are also studied and proved to be optimal.
I. INTRODUCTION
The paper studies a class of reducible cyclic codes over GF(p) with parity-check polynomial h1(x)h2(x), establishing three-weight behavior, weight distributions, and optimality results under specified parameter conditions.
- Research focus: The proposed cyclic code has length p^m−1 and parity-check polynomial h1(x)h2(x), with h1 and h2 derived from minimal polynomials over GF(p).The construction uses minimal polynomials associated with (−π)^−1 and π^−(p^k+1)/2.
- Research focus: The study determines when the code has exactly three nonzero weights, including the cases k/e odd or k even with e odd.Earlier results covered several special parameter settings, while this paper treats the broader stated conditions.
- Research focus: The paper determines the weight distributions of the proposed cyclic codes and studies the weight distributions of irreducible and reducible cyclic codes as established areas of research.The paper is organized around code construction, weight-distribution analysis, dual codes, and concluding applications.
- Research focus: Some constructed cyclic codes are optimal, and the duals of a subclass are also proved optimal.The paper motivates these constructions partly through potential applications in association schemes and secret sharing schemes.
II. QUADRATIC FORMS OVER FINITE FIELDS
This section introduces quadratic forms over finite fields as the mathematical framework used to calculate the cyclic codes’ weight distributions. It defines their coordinate representation, rank, diagonalization, and character-sum evaluation.
- Quadratic-form framework: A function from GF(q^s) to GF(q) is treated as a quadratic form when its representation on the s-dimensional GF(q)-vector space is homogeneous of degree two.A basis of GF(q^s) over GF(q) identifies field elements with vectors in GF(q)^s.
- Quadratic-form framework: The rank of a quadratic form is the codimension of its associated null space, whose size is q^(s−r) for rank r.This rank measures the dimension lost by the null space within the s-dimensional coordinate space.
- Quadratic-form framework: A quadratic form can be written as XAX′ with a symmetric matrix A and transformed by a nonsingular substitution into diagonal form.The nonzero diagonal coefficients determine the invariant η1(Δ), where Δ is their product.
- Evaluation tools: The paper uses standard quadratic-form lemmas to evaluate character sums according to the form’s rank and the parity of the rank and extension degree.The proof reduces the form by nonsingular substitution and distinguishes even-rank from odd-rank cases.
III. THE CLASS OF THREE-WEIGHT CYCLIC CODES AND THEIR WEIGHT DISTRIBUTION
The paper derives the value distributions underlying a class of three-weight cyclic codes in two parameter regimes and uses them to determine the codes’ weight distributions and parameters. Several resulting codes match best-known parameters or attain optimality bounds.
- Weight determination: Delsarte’s theorem expresses the codewords through exponential sums, whose value distributions determine the code’s weights.The paper separately defines S(a,b) and T(a,b) for the two parameter regimes.
- Value distributions: Quadratic-form analysis shows that S(a,b) takes only 0 and ±(p−1)p^(m+e)/2 when k is even and e is odd.The corresponding value frequencies are given explicitly in Theorem 3.4.
- Value distributions: When k/e is odd, T(a,b) takes only 0 and ±2(p−1)p^(m+e)/2, with the value distribution stated in Theorem 3.6.The calculation uses the possible ranks of the associated quadratic forms.
- Code parameters: For even k and odd e, C is a three-weight p-ary cyclic code with parameters [p^m−1,2m,p^m−p^(m−1)−p−1 2 p^((m+e−2)/2)].Its complete weight distribution is recorded in Table I.
- Code parameters: For odd k/e, C is a three-weight p-ary cyclic code with parameters [p^m−1,2m,p^m−p^(m−1)−(p−1)p^((m+e−2)/2)].Its complete weight distribution is recorded in Table II.
IV. THE DUALS OF A SUBCLASS OF THE CYCLIC CODES
The paper proves that a subclass of the cyclic codes has optimal ternary duals by establishing minimum distance four. The result includes general parameters and explicit optimal examples.
- The dual C⊥ is an optimal ternary code with parameters [3^m −1,3^m −1−2m,4] when p = 3, k is even, and e is odd.
- The proof excludes dual codewords of Hamming weights 2 and 3, while the Sphere Packing bound gives d ≤ 4.Therefore the minimum distance is four.
- The resulting subclass is optimal because its minimum distance is maximal for ternary linear codes with length 3^m −1 and dimension 3^m −1−2m.
- For m = 3, 5, and 7, the paper gives optimal ternary dual codes with parameters [26,20,4], [242,10,4], and [2186,14,4], respectively.The examples also specify generator polynomials.
V. SUMMARY AND CONCLUDING REMARKS
The paper presents a class of three-weight cyclic codes and determines their weight distributions. Some codes and the duals of a subclass are optimal, addressing the limited number of known three-weight constructions.
- The paper presents a class of three-weight cyclic codes and determines their weight distributions.
- Some of the constructed cyclic codes are optimal, and the duals of a subclass are also optimal.
- Because only a small number of three-weight codes are known, additional constructions are useful for association schemes and secret sharing schemes.