[Paper Review] Codes Associated with $O^+(2n,2^r)$ and Power Moments of Kloosterman Sums

Dae San Kim|arXiv (Cornell University)|Jul 29, 2008

Coding theory and cryptography15 references3 citations

TL;DR

This paper constructs three binary linear codes associated with orthogonal groups $SO^+(2,q)$, $O^+(2,q)$, and $SO^+(4,q)$ for $q = 2^r$, and derives recursive formulas for power moments of Kloosterman and 2D Kloosterman sums using the Pless power moment identity and Gauss sum expressions. The method yields a computationally more efficient approach than prior constructions based on $SL(2,q)$, with explicit examples demonstrating its effectiveness.

ABSTRACT

In this paper, we construct three binary linear codes $C(SO^+(2,q))$, $C(O^+(2,q))$, $C(SO^+(4,q))$, respectively associated with the orthogonal groups $SO^+(2,q)$, $O^+(2,q)$, $SO^+(4,q)$, with $q$ powers of two. Then we obtain recursive formulas for the power moments of Kloosterman and 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of Gauss sums for the orthogonal groups. We emphasize that, when the recursive formulas for the power moments of Kloosterman sums are compared, the present one is computationally more effective than the previous one constructed from the special linear group $SL(2,q)$. We illustrate our results with some examples.

Motivation & Objective

• To construct binary linear codes associated with the orthogonal groups $SO^+(2,q)$, $O^+(2,q)$, and $SO^+(4,q)$ for $q = 2^r$.
• To derive recursive formulas for power moments of Kloosterman and 2-dimensional Kloosterman sums using these codes.
• To improve computational efficiency in calculating power moments compared to previous methods based on $SL(2,q)$.
• To utilize explicit Gauss sum expressions for orthogonal groups to enable precise moment computations.
• To demonstrate the effectiveness of the proposed method through concrete examples.

Proposed method

• Construct three binary linear codes $C(SO^+(2,q))$, $C(O^+(2,q))$, and $C(SO^+(4,q))$ from the orthogonal groups $SO^+(2,q)$, $O^+(2,q)$, and $SO^+(4,q)$, respectively.
• Apply the Pless power moment identity to relate the power moments of Kloosterman sums to the weight distribution of the constructed codes.
• Utilize explicit expressions of Gauss sums for the orthogonal groups to compute the necessary moments.
• Derive recursive formulas for the power moments of Kloosterman and 2D Kloosterman sums based on the frequency of weights in the codes.
• Compare the computational efficiency of the new method with the prior approach based on $SL(2,q)$, showing improved performance.
• Illustrate the results with numerical examples to validate the recursive formulas and their practical utility.

Experimental results

Research questions

• RQ1 How can binary linear codes associated with orthogonal groups $SO^+(2,q)$, $O^+(2,q)$, and $SO^+(4,q)$ be constructed for $q = 2^r$?
• RQ2 What recursive formulas can be derived for the power moments of Kloosterman and 2D Kloosterman sums using these codes?
• RQ3 How does the computational efficiency of the new method compare to the previous construction based on $SL(2,q)$?
• RQ4 What role do explicit Gauss sum expressions for orthogonal groups play in enabling these moment computations?
• RQ5 In what ways do the weight distributions of the codes influence the derived moment formulas?

Key findings

• The paper successfully constructs three binary linear codes from the orthogonal groups $SO^+(2,q)$, $O^+(2,q)$, and $SO^+(4,q)$ for $q = 2^r$.
• Recursive formulas for the power moments of Kloosterman and 2-dimensional Kloosterman sums are derived using the Pless power moment identity and code weight distributions.
• The method based on orthogonal groups yields a computationally more efficient approach than the prior construction using $SL(2,q)$.
• Explicit Gauss sum expressions for the orthogonal groups are essential in enabling the derivation of the recursive formulas.
• The results are validated through illustrative examples, demonstrating the practicality and accuracy of the derived formulas.
• The framework provides a new pathway for computing power moments of exponential sums with improved efficiency over existing group-theoretic constructions.