Multi-Kernel Polar Code Generation by Column Permutation
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Solution Overview
Problem
The original construction of polar codes restricts code lengths to powers of 2, limiting their diversity and performance, especially in modern communication systems, and they have a limited range of minimum distances, which affects error rate performance.
Innovation Solution
A device and method for generating multi-kernel polar codes by permuting columns of transformation matrices to improve minimum distance and polarization properties without altering the decoding complexity, allowing for flexible code generation with lower error rates and reduced cyclic redundancy check size.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If original polar code construction is used, then code length is restricted to powers of 2, but this limits diversity and performance in modern communication systems
Solution Approach 1:
The transformation matrix is segmented into multiple kernels of different dimensions, where each kernel operates on a subset of bits. This segmentation allows the code length to be composed as a product of kernel dimensions (e.g., 24 = 4 × 6), breaking the restriction to pure powers of 2 while maintaining manageable construction complexity through modular design.
Solution Approach 2:
The invention introduces an additional dimension of flexibility by allowing kernels of different dimensions (not uniform 2×2 kernels). This dimensional variation in the transformation matrix enables code lengths that are products of different integers, significantly expanding the range of achievable code lengths beyond traditional power-of-2 constraints.
2Adaptability or versatility
If multi-kernel construction is used to achieve diverse code lengths, then code length flexibility improves, but minimum distance range remains limited
Solution Approach 1:
Different kernels in the multi-kernel construction are assigned different properties and dimensions tailored to specific requirements. By optimizing each kernel's contribution to the overall transformation, the system can achieve both diverse code lengths and improved minimum distance characteristics through localized optimization of kernel properties rather than uniform construction.
Solution Approach 2:
The transformation matrix is constructed as a composite of multiple kernels with different dimensions and properties. This composite structure combines the advantages of different kernel types, enabling the code to achieve both length flexibility and enhanced minimum distance by synthesizing multiple kernel contributions rather than relying on a single uniform kernel structure.
3Reliability
If column permutation of transformation matrix is applied, then minimum distance and reliability improve, but decoding complexity may increase
Solution Approach 1:
Column permutation of the transformation matrix is performed in advance during the code construction phase, before encoding and decoding operations. This preliminary arrangement optimizes the minimum distance and reliability properties of the code, while the permuted structure is then used consistently in both encoding and decoding, avoiding additional complexity during actual operation since the permutation is pre-baked into the transformation matrix.
Data Source
AI summary
A device for generating a multi-kernel polar code xN of length N and dimension K on the basis of a first transformation matrix GN of size N×N that defines a first multi-kernel polar code includes a processor configured to generate a second transformation matrix G′N of size N×N by permuting the order of at least two columns of a sub-matrix of the first transformation matrix GN, and generate the multi-kernel polar code xN an the basis of xN=uN·G′N, wherein uN=(u0, . . . , uN−1) is a vector of size N, with the elements ui, i=0, . . . N−1, corresponding to an information bit if i∈I, I being a set of K information bit indices, and ui=0, if i∈F, F being a set of N−K frozen bit indices.