| Summary: | Polar codes, which have been proposed as a family of linear block codes, has garnered a lot of attention from the scientific community, owing to their low-complexity implementation and provably capacity-achieving capability. Thus, they have been proposed to be used for encoding information on the control channels in the upcoming 5G wireless networks. The basic approach introduced by Arikan in his landmark paper to polarize bit channels of equal capacities to those of unequal capacities can be used to design only codewords of length <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><msup><mn>2</mn><mi>n</mi></msup></mrow></semantics></math></inline-formula>, which is a major limitation when codewords of different lengths are required for the underlying applications. In the predecessor paper, this aspect was partially addressed by using a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow></semantics></math></inline-formula> kernel circuit (used to generate codewords of length <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>=</mo><msup><mn>3</mn><mi>m</mi></msup></mrow></semantics></math></inline-formula>), along with downsizing techniques such as puncturing and shortening to asses the optimal design and resizing techniques based on the underlying system parameters. In this article, we extend this research to include the assessment of multi-kernel rate-matched polar codes for applicability over a much wider range of codeword lengths.
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