Systematic Code Decoding Using Low-Dimensional Erasure Equations
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Solution Overview
Problem
Current systematic code decoding methods for erasure channels are inefficient due to high-dimensional equations, requiring excessive computational time, especially when dealing with a large number of original packets, as seen in the decoding of Raptor codes, where the operation amount is substantial, taking approximately three and a half days for 10,000 iterations using a common personal computer.
Innovation Solution
The proposed method reduces decoding time by extracting a non-erased original packet from an encoded packet and using it to obtain an erased original packet through a lower-dimensional equation, significantly decreasing the operation amount required, thereby improving decoding efficiency, as demonstrated by reducing the decoding time for 10,000 iterations to approximately 22 minutes on a common PC.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If high-dimensional equations are used to decode systematic codes for erasure channels, then complete original packet sets can be recovered, but the decoding time becomes excessively long
Solution Approach 1:
The patent segments the decoding process into two distinct stages: first solving for intermediate packets using only received encoded packets, then solving for original packets using the intermediate packets. This segmentation reduces the computational complexity from handling all K original packets simultaneously to handling F erased packets separately, where F < K. The high-dimensional equation is divided into two lower-dimensional equations that can be solved more efficiently.
Solution Approach 2:
The patent extracts and solves for intermediate packets as a separate entity before recovering original packets. By introducing intermediate packets as a virtual representation, the method extracts the essential information from received encoded packets without directly solving the full high-dimensional system. This extraction approach reduces the dimensionality of the problem from K variables to F variables in the second stage.
2Quantity of substance
If the quantity of original packets K is increased to handle larger data sets, then the encoding capacity improves, but the decoding operation amount increases significantly
Solution Approach 1:
The patent segments the decoding task into two phases where the computational burden is distributed. The first phase processes received encoded packets to obtain intermediate packets, and the second phase uses these intermediates to recover only the F erased original packets. This segmentation ensures that the operation amount scales with F (number of erasures) rather than K (total original packets), enabling efficient decoding even when K is very large.
Solution Approach 2:
The patent changes the parameter being solved for from direct original packets to intermediate packets in the first stage. This parameter transformation allows the system to work with a reduced dimensionality representation. The intermediate packets serve as a compressed representation that captures the essential information needed to recover original packets, changing the problem from solving K variables to solving F variables.
3Ease of manufacture
If direct matrix multiplication is performed on intermediate packets to obtain original packets, then the decoding process follows the standard algorithm, but the computational complexity becomes unmanageable for large K
Solution Approach 1:
The patent segments the matrix operations into two separate matrix multiplications rather than one large operation. The first multiplication computes intermediate packets from received encoded packets, and the second multiplication recovers original packets from intermediates. This segmentation reduces the computational complexity from O(K²) to O(F² + K·F), making the process manageable for large K while maintaining algorithmic simplicity.
Data Source
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AI summary
Embodiments of the present invention disclose a systematic code decoding method and an apparatus, so as to greatly reduce an operation amount of decoding and save decoding time. The method in the embodiments of the present invention includes: first extracting, by a decoding apparatus, a non-erased original packet from an acquired encoded packet that has passed through an erasure channel; and then by using the erased original packet as an unknown variable, obtaining an erased original packet by parsing according to a first equation, where the first equation is an equation indicating a correspondence between the erased original packet and the acquired encoded packet. Because a quantity F of erased original packets is far less than a total quantity K of original packets, a dimension of the first equation that includes F unknown variables is far lower than a dimension of an equation indicating a correspondence between an encoded packet and an intermediate packet. An operation amount required for parsing the relatively low-dimensional first equation is far smaller than that for parsing the quite high-dimensional equation indicating the correspondence between an encoded packet and an intermediate packet, thereby greatly reducing an operation amount required for decoding.