A Multi-Dimensional Parity Approach for Error Correction in Data Systems

Author:
Farid Zehetbauer

Abstract:
This document outlines a novel method for error correction that extends traditional 2D parity checks to three or more dimensions. By applying parity checks in multiple dimensions, this technique offers enhanced detection and correction capabilities for multiple bit errors, suitable for both classical and quantum computing environments.

1. Introduction

Error correction in data systems typically involves techniques like Reed-Solomon codes or Low-Density Parity-Check (LDPC) codes. However, these methods may not suffice in environments with very high error rates, such as quantum computing. This paper introduces an approach where parity checks are computed across multiple dimensions to achieve better error correction.

2. Concept of Multi-Dimensional Parity

2D Parity:

• Grid Model: 1 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1
  • Here, the last column is the parity for each row, and the last row is the parity for each column. The bottom-right bit (1) is the parity of all these parity bits.

3D Extension:

• Parity is calculated for each dimension (x, y, z) of a data cube.
• A parity bit for the parity bits is included (e.g., at the corner of a 3D grid).

N-Dimensional Parity:

• Data is conceptualized in an N-dimensional space with parity checks for each dimension.
• Description of how errors can be detected and corrected in such a system.

3. Application in Quantum Computing

• High error rates in quantum systems necessitate advanced error correction.
• This method could potentially reduce the number of physical qubits needed for error correction by leveraging multi-dimensional parity.

4. Methodology

• Description of how data would be encoded in multi-dimensional space.

• Basic outline of an algorithm to detect and correct errors based on parity failures:

  1. Encoding: Place data bits in a multi-dimensional grid, add parity bits for each dimension.
  2. Error Detection: Check parity in each dimension. Errors manifest as parity mismatches.
  3. Error Correction: Use the pattern of parity failures to locate and correct errors.

5. Conclusion

This multi-dimensional parity approach introduces a new perspective on error correction, potentially applicable to both classical and quantum data systems. By making this concept public, we aim to enrich the field's knowledge base and ensure it remains open for further development.

References
Q. L. Rao, C. He (2009). A new 2-D parity checking architecture for radiation-hardened by design SRAM. Asia Pacific Conference on Postgraduate Research in Microelectronics & Electronics. pp. 360–363. J. M. Shea, T. F. Wong (2003). "Multidimensional Codes". Encyclopedia of Telecommunications. Wiley. Ludek Dudácek, Ivo Vertat (2016). Multidimensional Parity Check codes with short block lengths. 24th Telecommunications Forum TELFOR. pp. 1–4. A. Vadinala, G. K. Kumar (2013). Multi Dimensional Parity Based Hamming Codes For Correcting The SRAM Memory Faults Under High EMI Conditions. IACEECE International Conference. pp. 46–49.

Date:
2025-01-25