Turán-Theoretic Bounds on Several Elementary Trapping Sets in LDPC Codes
For coding theorists, this work provides theoretical bounds on trapping sets that can help design LDPC codes with lower error floors.
This paper uses Turán numbers to derive lower bounds on the size of elementary trapping sets (ETSs) in LDPC codes, showing that the minimum ETS size increases significantly under certain conditions. The results are validated by constructing QC-LDPC codes with improved error floor performance.
LDPC codes have attracted significant attention because of their superior performance close to the Shannon limit. Elementary trapping sets are the main cause of the error floor phenomenon in LDPC codes. We consider typical graphs related to trapping sets, including theta graphs, dumbbell graphs, and short cycles with chords. Based on the Turán numbers of $θ(2,2,2)$, $θ(1,3,3)$ and $D(4,4;0)$, we prove that any $(a,b)$-ETS with $g=8$ variable-regular $γ$ satisfies the inequality $b\geq aγ-\frac{a(\sqrt{24a-23}-1)}{4}$, provided that any two 8-cycles in the Tanner graph do not share common variable node. In addition, we can also eliminate ETSs by removing certain short-cycle structures with chords. The minimum sizes of ETSs obtained through these methods are significantly increased. To assess practical impact , we analyze spectral radii of the ETSs and construct QC-LDPC codes to show frame error rates in the error floor region.