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This paper gives necessary and sufficient conditions for the Tanner graph ofa quasi-cyclic (QC) low-density parity-check (LDPC) code based on the all-oneprotograph to have girth 6, 8, 10, and 12, respectively, in the case ofparity-check matrices with column weight 4. These results are a naturalextension of the girth results of the already-studied cases of column weight 2and 3, and it is based on the connection between the girth of a Tanner graphgiven by a parity-check matrix and the properties of powers of the productbetween the matrix and its transpose. The girth conditions can be easilyincorporated into fast algorithms that construct codes of desired girth between6 and 12; our own algorithms are presented for each girth, together withconstructions obtained from them and corresponding computer simulations. Moreimportantly, this paper emphasizes how the girth conditions of the Tanner graphcorresponding to a parity-check matrix composed of circulants relate to thematrix obtained by adding (over the integers) the circulant columns of theparity-check matrix. In particular, we show that imposing girth conditions on aparity-check matrix is equivalent to imposing conditions on a square circulantsubmatrix of size 4 obtained from it.

Necessary and Sufficient Girth Conditions for LDPC Tanner Graphs with Denser Protographs