We present a tree-based construction of LDPC codes that have minimumpseudocodeword weight equal to or almost equal to the minimum distance, andperform well with iterative decoding. The construction involves enumerating a$d$-regular tree for a fixed number of layers and employing a connectionalgorithm based on permutations or mutually orthogonal Latin squares to closethe tree. Methods are presented for degrees $d=p^s$ and $d = p^s+1$, for $p$ aprime. One class corresponds to the well-known finite-geometry and finitegeneralized quadrangle LDPC codes; the other codes presented are new. We alsopresent some bounds on pseudocodeword weight for $p$-ary LDPC codes. Treatingthese codes as $p$-ary LDPC codes rather than binary LDPC codes improves theirrates, minimum distances, and pseudocodeword weights, thereby giving a newimportance to the finite geometry LDPC codes where $p > 2$.
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机译：我们提出了一种LDPC码的树结构，其最小伪码字权重等于或几乎等于最小距离，并且在迭代解码中表现良好。构造涉及枚举固定数目的层的规则树，并使用基于置换或相互正交的拉丁方的连接算法来关闭树。给出了度数为$ d的质数$ d = p ^ s $和$ d = p ^ s + 1 $的方法。一类对应于众所周知的有限几何和有限广义四边形LDPC码。其他提供的代码是新的。我们还给出了$ p $ ary LDPC码的伪码字权重的一些界限。将这些代码视为$ p $ ary LDPC代码而不是二进制LDPC代码可提高其速率，最小距离和伪代码字权重，从而使$ p> 2 $的有限几何LDPC代码具有新的重要性。
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