A Novel Computational Framework for Weight Distributions in QR Codes: The Case of (57,29,17)

Abstract

This study presents a novel methodology for deriving both the received polynomial and the error locator polynomial for the Quadratic Residue (QR) code (57,29,17). Utilizing a newly developed algorithm, the weight distribution of this QR code is efficiently computed through Magma computations. Furthermore, the study introduces an innovative algorithm specifically de- signed to calculate Lee weights for the QR code, enhancing the accuracy and efficiency of these computations. The proposed techniques significantly contribute to the field of coding theory by improving the decoding process and providing deeper insights into the properties of QR codes.

Objectives: The paper presents a novel computational framework for analyzing the Quadratic Residue (QR) code (57,29,17). To compute the weight distribution of the QR code using an efficient algorithm implemented in Magma. To derive the Lee weight distribution of the QR code using a newly developed algorithm implemented in SageMath. To construct and analyze the error locator polynomial, which aids in error detection and correction for the QR code.

Methods: Constructed the generator matrix using quadratic residues modulo 57. Generated codewords by taking linear combinations of rows in the matrix. Computed the Hamming weight of each codeword. Counted occurrences of different weights to form the weight distribution table. Defined the Lee weight metric for elements in ℤ₅₇​. Randomly generated codewords and calculated their Lee weights using parallel computing. Aggregated results to derive the Lee weight distribution of the QR code. Considered cases of 1 to 8 errors in the received polynomial. Derived the error locator polynomial for each case. Demonstrated the process of recovering the transmitted message using syndromes.

Results: Successfully computed and tabulated the weight distribution of (57,29,17). Showed how different weights contribute to the structure and error-correcting capability of the code. Derived the Lee weight distribution, providing a new perspective for analyzing the QR code. Demonstrated that the Lee weight metric can be effectively applied to QR codes over finite rings. Derived error locator polynomials for different error cases. Confirmed that the QR code (57,29,17) can detect and correct up to 8 errors.

Conclusions: In this study focuses on the in-depth analysis of the Quadratic Residue code (57, 29, 17) by deriving both its weight distribution and Lee weight distribution. The newly developed algorithms for calculating these distributions contribute significantly to understanding the code’s structure and its error detection and correction capabilities. Additionally, the error locator polynomial for the code has been successfully found, which further enhances its practical application in error correction