# The Cyclically Resolvable Steiner Triple Systems of Order 57
## 1 Introduction  
**Definition 1**. : _A Steiner triple system of order \(v\) (STS(\(v\))) is a pair \(D=(V,B)\) where \(V\) is a set of \(v\) elements called points and \(B\) a family of 3-element subsets (triples) of \(V\) called blocks, such that each pair of elements of \(V\) is contained in exactly one block of \(B\)._  
An STS(\(v\)) has \(b=v(v-1)/6\) blocks and each of its points is in \(r=(v-1)/2\) blocks. Its incidence matrix has \(v\) rows and \(b\) columns and the element in the \(i\)-th row and \(j\)-th column is 1 if point \(i\) is in block \(j\), or 0 otherwise. An STS(\(v\)) is a 2-\((v,3,1)\) block design. That is why we will further talk either of STS(57)s, or of 2-\((57,3,1)\) designs. Two STS(\(v\))s are isomorphic if there is a bijection of the point sets such that blocks are mapped to blocks. An automorphism of an STS(\(v\)) is an isomorphism to itself. All the automorphisms of an STS(\(v\)) form its full automorphism group. A parallel class of an STS(\(v\)) is a set of \(v/3\) blocks, such that each point is in exactly one block. A resolution of an STS(\(v\)) is a partition of the collection of blocks to parallel classes. Two resolutions \(R\) and \(R^{\prime}\) are isomorphic if there exists an automorphism of the underlying STS(\(v\)) which maps each parallel class of \(R\) to a parallel class of \(R^{\prime}\). An automorphism of the resolution \(R\) is an automorphism of the STS(\(v\)) which maps each parallel class of \(R\) to a parallel class of \(R\). An STS(\(v\)) together with a resolution of its blocks is called a Kirkman triple system (KTS(\(v\))). For the basic knowledge on STS(\(v\))s and KTS(\(v\))s refer, for instance, to [1, 2]. KTS(\(v\))s are related to error correcting codes in many different ways [3]. It is well known that a KTS(\(v\)) is equivalent to an optimal \((r,v,r-1)_{v/3}\) equidistant code (length \(r\), \(v\) codewords, distance \(r-1\), alphabet of \(v/3\) elements) [4]. An STS(\(v\)) is cyclic if it has an automorphism of order \(v\) permuting its points in one cycle. A cyclic STS(\(v\)) corresponds to a \((v,3,1)\) cyclic difference family. A resolution is point-cyclic if it has an automorphism permuting the points in one cycle. An STS(\(v\)) is resolvable if it has at least one resolution, and it is cyclically resolvable if at least one of its resolutions is point-cyclic. **Example 1**. : _A cyclic STS(\(15\)) is presented in Figure 1. It is invariant under an automorphism of order \(15\) which permutes the points in one cycle. It generates a cyclic group of order \(15\) which partitions the blocks to three orbits, namely one short orbit of length \(5\) and two full orbits of length \(15\). The full automorphism group of this design is of order \(20{,}160\). One of its resolutions is presented in the figure too. It has seven parallel classes, the short block orbit being one of them. This resolution is invariant under an automorphism group of order \(168\) which is transitive on the parallel classes, but it is not invariant under the cyclic group of order \(15\) and is therefore not point-cyclic._  
The incidence matrix of an STS(\(v\)) can be used as a \(v\times b\) parity-check matrix of a binary \((3,r)\)-regular Low-Density Parity-Check (LDPC) code with \(v\) parity-check equations, \(b\) codeword bits, row weight \(r\) and column weight \(3\)[5]. If the STS(\(v\)) is cyclic, the parity  
Figure 1: Cyclic \(2\)-\((15,3,1)\) design = cyclic STS(\(15\)), \(v=15\) points, \(b=35\) blocks, \(r=7\). check matrix is quasi-cyclic (contains circulant submatrices) and therefore allows linear-complexity encoding with simple feedback shift registers [6; 7; 8]. KTS(\(v\))s are useful when the row weight of such an LDPC code has to be reduced by removing some columns of the parity-check matrix in such a way that the same number of ones are cut off from each row (true if one or several parallel classes are removed) to obtain a lower rate code without changing the other code properties [5; 6; 9; 10].