An optimized detection algorithm for identifying 6-cycles in quasi-cyclic polar codes

Abstract

This paper focuses on 6-cycles, a type of short cycles that degrade error-correction performance in quasi-cyclic (QC) polar codes. This is a class of polar codes that achieve significant coding gain, reduced complexity, and high throughput through a protograph-based structure. The 6-cycles are known to harm decoding efficiency in codes graph. We analyze their structural property and key characteristics of this code. Additionally, we propose a novel optimization-based detection method that efficiently identifies these short cycles.

A major contribution of our work is the introduction of a direct connection matrix (D), which is a representation of the code’s graph structure that simplifies the cycle detection. Indeed, this matrix significantly reduces the search space for finding 6-cycles, making the detection process more computationally efficient. Furthermore, we find symmetric and reflection-symmetric structures in QC polar codes. These structures reduce the search complexity to one-fourth of the original search complexity, drastically lowering computational overhead.

Finally, we investigate the distribution of 6-cycles in codes graph of different lengths. Then, via a recursive formula, we examine the number of 6-cycles appearing in polar codes of different lengths.