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Cages, defined as regular graphs with minimum number of nodes for a givengirth, are well-studied in graph theory. Trapping sets are graphical structuresresponsible for error floor of low-density parity-check (LDPC) codes, and arewell investigated in coding theory. In this paper, we make connections betweencages and trapping sets. In particular, starting from a cage (or a modifiedcage), we construct a trapping set in multiple steps. Based on the connectionbetween cages and trapping sets, we then use the available results in graphtheory on cages and derive tight upper bounds on the size of the smallesttrapping sets for variable-regular LDPC codes with a given variable degree andgirth. The derived upper bounds in many cases meet the best known lower boundsand thus provide the actual size of the smallest trapping sets. Consideringthat non-zero codewords are a special case of trapping sets, we also derivetight upper bounds on the minimum weight of such codewords, i.e., the minimumdistance, of variable-regular LDPC codes as a function of variable degree andgirth.

From Cages to Trapping Sets and Codewords: A Technique to Derive Tight  Upper Bounds on the Minimum Size of Trapping Sets and Minimum Distance of  LDPC Codes