Using Partial Orthomorphisms to Construct Short Quasi-Cyclic LDPC Codes with Girth at Least 6

Since the publication of Fossorier’s work on the necessary and sufficient conditions for a $(k,\ell)$-regular QC-LDPC code to have a desired girth g, many researchers have been attracted to the problem of determining the smallest lifting degree $N_{\text{min}}$ for which such a code exists. For some values of $ k,\ell$ and g, either an explicit algebraic formula for $N_{\min}$ has been given or its value has been determined by exhaustive computer search. For most cases, however, only lower bounds are known. In this paper, we translate the problem of determining $N_{\min}$ into one about the existence of mutually adjacent partial orthomorphisms of $\mathbb{Z}_{N}$ with certain deficit d, and determine its value for certain parameters. These particular orthomorphisms can also be used to determine, for a fixed $\ell$ and N, the maximum number $k_{\max}$ for which there exists a $(k_{\max},\ell)$-regular QC-LDPC code with girth g.