Reed-Solomon Encoding With Vandermonde Order Reduction
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Solution Overview
Problem
Conventional error correcting coding techniques have high processing amounts, particularly in field multiplication during the encoding process.
Innovation Solution
A matrix application method that computes vector b by multiplying vector a with a Vandermonde matrix, incorporating polynomial multiplication and order reduction to reduce the processing amount of field multiplication, using a polynomial h obtained by dividing terms of order q or higher by Xq< and forming polynomial g from terms of order lower than q.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If conventional error correcting coding techniques are used, then error correction capability is achieved, but processing amount is high
Solution Approach 1:
The patent segments the field multiplication operation into two distinct parts: (1) multiplication of terms with order less than q, and (2) multiplication of terms with order q or higher. This segmentation allows different processing strategies to be applied to each part, reducing the overall processing amount while maintaining error correction capability.
Solution Approach 2:
The patent introduces a parameter q (where q < m) that divides the polynomial terms into two groups based on their order. By changing the parameter threshold for processing, the patent optimizes the computational complexity of field multiplication while preserving the mathematical properties required for Reed-Solomon error correction.
2Ease of manufacture
If field multiplication is performed in the encoding process, then encoding is completed, but processing complexity increases
Solution Approach 1:
The encoding process is segmented into distinct computational paths: one for low-order terms (order < q) and another for high-order terms (order ≥ q). This segmentation simplifies the overall processing complexity by allowing each segment to be handled with optimized, specialized operations rather than a single complex multiplication routine.
Solution Approach 2:
The patent uses an intermediary polynomial representation where high-order terms are processed separately and then combined with low-order terms. This intermediary step acts as a mediator that reduces the direct computational burden between the input message and the final encoded output, thereby reducing processing complexity.
Data Source
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AI summary
To reduce the processing amount of a field multiplication. A matrix application apparatus 1 computes a vector b by multiplying a vector a and a matrix A, provided that a denotes a k-th order vector having elements a0, ..., ak-1 (a0, ..., ak-1∈GF(xq)), b denotes an m-th order vector having elements b0, ..., bm-1 (b0,..., bm-1∈GF(xq)), and A denotes a m-by-k Vandermonde matrix. A polynomial multiplication part 12 computes a value bi. An order reduction part 13 designates gi-hif' as the value bi by using a polynomial hi obtained by dividing a part of the value bi having an order equal to or higher than q by Xq and a polynomial gi formed by a part of the value bi having an order lower than q.