On the Girth of Tanner QC-LDPC Cycle Codes: An Algebraic Number Theory Approach

Tanner quasi-cyclic low-density parity-check (QC-LDPC) codes form an important family of structured LDPC codes with favorable girth properties. This paper studies the girth of Tanner (2, L)-regular QC-LDPC codes (referred to as Tanner QC-LDPC cycle codes) for arbitrary integers L>2 and develops a novel algebraic number theoretic method to determine the girth for all sufficiently large primes p with p≡1(mod2L). We first analyze the case L=3 and prove that the girth is 12 for every prime p≡1(mod6) through exhaustive resultant computations. We then extend the method to arbitrary L and obtain a clear classification: when L is even, the girth is exactly 8 for all admissible primes; when L is odd, the girth attains the maximum value 12 for all sufficiently large admissible primes. The proof transforms cycle existence conditions into polynomial equations and applies resultant theory. This approach converts the infinite task of checking all primes into a finite set of algebraic checks. Numerical simulations show that the Tanner (2, 5)-regular non-binary code over GF(64) achieves a coding gain of approximately 0.2 dB over the 5G LDPC code of equivalent binary length.