On the Girth of Tanner (3, 13) Quasi-Cyclic LDPC Codes

Abstract

Girth is an important structural property of low-density parity-check (LDPC) codes. Motivated by the works on the girth of Tanner (3, 5), (3, 7), (3, 11), and (5, 7) quasi-cyclic (QC) LDPC codes, we, in this paper, study the girth of Tanner (3, 13) QC-LDPC codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$13p$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> being a prime of the form <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(39m+1)$ </tex-math></inline-formula> . First, the cycle structure of Tanner (3, 13) QC-LDPC codes is analyzed, and the cycles of length lesser than 12 are divided into five equivalent classes. Based on each equivalent class, the existence of the cycles is equivalent to the solution of polynomial equations in a 39th unit root in the prime filed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{p}$ </tex-math></inline-formula> . By solving these polynomial equations over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{p}$ </tex-math></inline-formula> and summarizing the resulting candidate prime values, the girth of Tanner (3, 13) QC-LDPC codes is obtained. As an advantage, Tanner (3, 13) QC-LDPC codes have much higher code rates than Tanner (3, 5), (3, 7), (3, 11), and (5, 7) QC-LDPC codes, and provide a promising coding scheme for the data storage systems and optical communications.