# The Cyclically Resolvable Steiner Triple Systems of Order 57
## 4 Results and Open Problems  
We construct all point-cyclic KTS(57)s. The number of automorphisms of the cyclically resolvable STS(57)s and of their resolutions is presented in Table 5, where the abbreviation \(D\) stands for _designs_ (STS(57)s), \(R\) for _point-cyclic resolution_, \(AutD\) denotes the number of automorphisms of the designs, and \(AutR\) of the resolutions. It can be seen that less than 7% of all cyclic STS(57)s are cyclically resolvable and most of their resolutions possess no automorphisms except those from the constructive automorphism group \(G_{57}\). There are 1592 point-cyclic KTS(57)s with a full automorphism group of order 171 and 2 with a full automorphism group of order 513. Some of the point-cyclic KTS(57)s with a full automorphism group of order 171 have only two parallel class orbits under their full automorphism group (Table 6). In this case there is an automorphism of order three which maps to one-another the parallel class orbits of length 3 under \(G_{57}\), and this way these 9 classes form one class orbit under the full automorphism group of the resolution. Each cyclically resolvable STS(57) has between 1 and 154 resolutions. The distribution of the number of point-cyclic resolutions of an STS(57) is presented in Figure 2. The maximal \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**AutD** & **171** & \multicolumn{2}{c}{**342**} & **1026** & **6498** & **All** \\ \hline parralell class orbits & 2 & 4 & 2 & 4 & 2 & 4 & - \\ point-cyclic KTS(57)s & 170 & 1350 & 36 & 24 & 1 & 11 & 1592 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parallel class orbits of point-cyclic KTS(57)s with a full automorphism group of order 171. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**AutD** & **57** & **114** & **171** & **342** & **513** & **1026** & **1083** & **6498** & **19,494** & **All** \\ \hline STS(57)s & 2,351,358 & 74 & 1834 & 33 & 4 & 3 & 1 & 2 & 1 & 2,353,310 \\ cycl. resolvable D & 154,857 & 46 & 1027 & 31 & 0 & 1 & 1 & 2 & 1 & 155,966 \\ \hline R with AutR = 57 & 3,618,667 & 2342 & 15,669 & 508 & 0 & 6 & 117 & 75 & 6 & 3,637,390 \\ R with AutR = 171 & 0 & 0 & 1520 & 60 & 0 & 1 & 0 & 11 & 0 & 1592 \\ R with AutR = 513 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 2 \\ \hline All R & 3,618,667 & 2342 & 17,189 & 568 & 0 & 8 & 117 & 86 & 7 & 3,638,984 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Automorphisms of the cyclically resolvable STS(57)s and their resolutions. number 154 of point-cyclic KTS(57)s of one cyclic STS(57) is achieved by an STS(57) with a full automorphism group \(G_{57}\). There are \(11,857\) STS(57)s with 21 point-cyclic resolutions. A full \(m\)-configuration in an STS(\(v\)) is a set of \(m\) blocks which cover \(m+2\) points. Each point is in at least two blocks. Full configurations on 4 and 5 blocks are unique and also known as the Pasch and the mitre configurations. The absence of these configurations in a cyclically resolvable cyclic STS(\(v\)) is a desirable property with respect to coding applications [6; 7; 10; 30]. The minimum distance of an LDPC code is equal to the minimum nonzero number of columns in its parity-check matrix for which a nontrivial linear combination sums to zero (mod 2) [31]. For an LDPC code from an STS(\(v\)) the minimum distance constructively is settled to at least 4 (each column has weight 3 and two columns share no more than one point). The 4 blocks of a Pasch configuration sum to zero (mod 2), while the 5 blocks of a mitre configuration do not. In _anti-Pasch_ STS(\(v\))s no four triples are isomorphic to the Pasch configuration. That is why an LDPC code from an anti-Pasch STS(\(v\)) has minimum distance at least 6 [5]. In _anti-mitre_ STS(\(v\))s no five triples are isomorphic to the mitre configuration. If an STS(\(v\)) is without any Pasch or mitre configuration it is 5-_sparse_. 5-sparse KTS(\(v\))s are of major importance not only for the constructions of LDPC codes, but for some other applications in Coding Theory too.