MDS Erasure Decoding with Sub-Matrix Inversion to Cut Field Multiplication
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Solution Overview
Problem
Conventional error correcting coding techniques face high processing amounts, particularly in field multiplication during the decoding process.
Innovation Solution
A decoding apparatus and method that reduces processing amounts by generating a φ-th order vector and φ-by-φ inverse matrix, allowing for efficient computation of plaintext elements using specific formulas and matrix operations.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional decoding methods using full matrix inversion are used, then decoding accuracy is maintained, but processing amount of field multiplication becomes excessively high
Solution Approach 1:
The patent segments the decoding process by dividing the received vector into systematic parts and redundant parts, and by partitioning the inverse matrix computation into smaller sub-matrices. This segmentation allows selective processing of only necessary elements rather than computing the full inverse matrix, thereby reducing field multiplication operations while maintaining decoding accuracy.
Solution Approach 2:
The patent extracts and utilizes only the necessary rows and columns from the full inverse matrix based on the pattern of lost packets. By taking out only the relevant sub-matrix elements required for recovering specific lost data, the computation is significantly reduced while still achieving accurate decoding of the lost packets.
2Reliability
If systematic MDS codes with Vandermonde matrices are used, then erasure recovery capability is improved, but decoding complexity increases due to matrix inversion requirements
Solution Approach 1:
The patent applies local quality by making the decoding complexity adaptive to the specific erasure pattern. Instead of always performing full matrix inversion, the method adjusts the computation to process only the local sub-matrix corresponding to the actual lost packets, thereby reducing overall decoding complexity while maintaining the erasure recovery capability of systematic MDS codes.
Solution Approach 2:
The patent changes the parameter of matrix size from the full n×n inverse matrix to a smaller φ×φ sub-matrix, where φ is the number of lost packets. This parameter change dynamically adapts the computation to the actual erasure scenario, reducing decoding complexity while preserving the reliability of erasure recovery.
Data Source
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AI summary
To reduce the processing amount of a field multiplication. a denotes a k-th order vector whose elements are a0, ..., ak-1 (a0, ..., ak-1∈GF(xq)). A denotes an n-by-k matrix formed by vertically connecting a identity matrix and a Vandermonde matrix. b denotes an n-th order vector obtained by multiplying the vector a and the matrix A whose elements are b0, ..., bn-1 (b0, ..., bn-1∈GF(xq)). A vector conversion part 11 generates a φ-th order vector b' using φ elements bp0, ..., bpφ-1 of the vector b. An inverse matrix generation part 12 generates a φ-by-φ inverse matrix A'-1. A plaintext computation part 13 computes elements ae0, ..., aeφ-1 of the vector a by multiplying the vector b' and the inverse matrix A'-1.