This directory contains the data for the translation planes of order 49. The following files are present: data49 - the MASTER file of data for the translation planes. This is the version of the datafile that I am now happy with, and would like to see updated if at all possible. If you refer to translation planes from this work, please use the numbering as in this file. finaldata - the data regarding the translation planes this is the same data (minus the fingerprints) but in a less useful format. linespg37 - the lines of PG(3,7) linmatpg37 - index from line numbers to 2 x 2 matrices trans49.tex LaTeX source for the paper on translation planes trans49.dvi .dvi version of the above trans49.ps Postscript version of the above Notes: American users will wish to edit trans49.tex to eliminate the inclusion of the style file a4.sty which is designed for A4 sized paper. The document trans49.tex now supercedes the original technical report. ERRATA NOTICE: The original technical report dated 4/93 contains a typographical error in the Examples section in that the symbols for W_0 and W_infinity were consistently swapped. This only affects the PRINTED version of the technical report, the data files are correct, but the program that took a data file and produced TeX input was in error. Even then this only affects the orbit partitions of the spreads. The file data49 --------------- This contains 1347 sets of 10 records in the following format N> 1 R> 0 0 0 0 690 16840 F> 970 320 440 480 240 0 0 0 0 0 0 0 0 G> 20 K> 935 B> 5 (2) 20 (2) P> 0 777 800 860 965 1017 1043 1128 1362 1420 1454 1606 1773 1914 2087 2383 2448 2583 2735 2811 | 449 505 561 680 737 926 1147 1201 1491 1542 1666 1722 1957 19 96 2022 2209 2314 2652 2670 2789 | 618 1237 2282 2491 2509 | 1328 1747 1839 21 24 2224 | S> 7777 4606 1110 2511 3513 4114 0615 6016 4324 5525 3426 4232 0136 6241 3045 32 54 5455 3661 4464 1366 | 0000 1001 2002 5004 6105 5112 1520 2321 1630 2131 5633 6634 5342 4043 0544 6350 0353 6562 2263 5265 | 3103 0422 2652 4556 0260 | 6423 3 335 2440 1246 1451 | RC> dbe0376bda768709 f56b07e6313af59b FC> e3070710102d48fd ac7285b9643559f3 The fields all commence with a code followed by a >. This is to allow easy access to any particular item of data desired. The codes have the following meanings N> The number of the plane. The numbers range from 1 .. 973 with polar pairs being distinguished by suffixing "a" and "b" onto their numbers. Hence it is clear that 940a and 940b form a polar pair, whereas number 1 for example has an isomorphic polar mate. R> The short regulus profile of the spread. The entries form a vector x_8 x_7 .... x_3 where x_i is the number of reguli that intersect the spread S in i points. F> The Conway/Charnes (*) fingerprint of the spread. The fingerprint is computed from the spread set as follows (see below for spread set). A 49x49 matrix is formed with (i,j) entry equal to 0, +1, -1 according as the determinant of A_i - A_j is 0, square or non-square. This matrix is then bordered with 0111111 as a leading row and column, the resulting 50x50 symmetric matrix called A. Then the matrix Q = AA^t is formed. The multiset of the absolute values of the entries of this matrix is the fingerprint. In fact our fingerprint lists x_0 x_4 x_8 ... x_48 where x_i is the number of occurrences of +i or -i in Q. For these planes no other entries occur except 49 which always appears 50 times, so is ignored. (*) This invariant was first discovered and exploited by Conway and Charnes, but was expressed in terms of the plane. Moorhouse first described to me how to compute it directly from the spread set. G> The order of the stabiliser group of the spread in PGammaL(4,7). Computed as described in the paper. K> The 7-rank of the incidence matrix of the translation plane. Computed using sparse-matrix techniques on the 2451x2451 incidence matrix of the actual translation plane. B> The orbit structure of the stabiliser of the spread acting on the spread itself. The listing gives the number of orbits of each size. For example 5 (2) 20 (2) indicates two orbits of length 5 and two of length 20. P> The actual spread itself, given in terms of LINE NUMBERS. The line numbers are potentially arbitrary, but the actual lines are given in the file linespg37 which is also in this directory. The spread is also given orbit by orbit, with each orbit separated by a |. S> The spread itself, given now in terms of the spread-sets. Recall that a spread is really a set of two-dimensioanl subspaces of GF(7)^4. If we assume that one of the subspaces is W_inf = {(0,0,x,y) | x,y in GF(7)} and another is W_0 = {(x,y,0,0) | x,y in GF(7)} then all the others can be described by a 2x2 non-singular matrix A as W_A = {(x,xA) | x in GF(7)^2}. Therefore the set consisting of 49 matrices with the first being 0 and the remainder non-singular can describe a spread. We use a special symbol 7777 to refer to W_inf and the remaining matrices are given row-by-row. Hence an orbit 7777 1001 2341 0000 would mean W_inf and W_0 together with the two matrices 1 0 2 3 0 1 and 4 1 This gives the spread set - in the same orbit order as previously. Notice that in any polar pair (for example 940a and 940b) the two spread sets will be the transposes of each other. The corresponding orbits will be listed in the same order, but within orbits the elements may not be listed in the same order. RC> FC> These two are technical internal codes used for distinguishing spreads. A casual user will have no use for them. For a programmer the details follow: They are the hash codes produced by the naututil.c commands hash(canong,m*n,7), hash(canong,m*n,13) when applied to the canonically labelled point/line graph of PG(3,7) with the spread stabilised. They are hexadecimal versions of 64-bit integers (hence must be produced on a 64-bit machine, such as a Dec Alpha which was what produced these). To obtain these numbers nauty must be called with the lines partitioned into cells, the cells ordered either with the regulus partitioning (to get RC) or with the fingerprint partitioning (to get FC). If the cells are *not* ordered in the same fashion then the codes may be different, and hence useless. The points of PG(3,7) must appear in the first cell, then the lines of the spread partitioned according to the profiles, and finally the remainining lines. Future additions Any additions to this file can clearly be easily made simply by selecting a new code and interleaving the results with the current one. Any user who computes any further data is encouraged to submit them to me (gordon@cs.uwa.edu.au) for incorporation. For example, one might like to consider the number of Fano planes through a point. I can do it, using about an hour on a Dec Alpha for each plane - unfortunately I don't have 1347 Alpha hours free. If anyone has any free cpu time and has a sysadmin willing to let me use it, then let me know :-). The file finaldata ------------------ This file contains 1347 entries of the following form: 1. profile= 0 0 0 0 690 16840, group= 20, rank=935, paired=no, orbs= 5 (2) 20 (2) 1328 1747 1839 2124 2224 | 618 1237 2282 2491 2509 | 0 777 800 860 965 1017 1043 1128 1362 1420 1454 1606 1773 1914 2087 2383 2448 2583 2735 2811 | 449 505 561 680 737 926 1147 1201 1491 1542 1666 1722 1957 1996 2022 2209 2314 2652 2670 2789 | 1. indicates that this is the first translation plane profile gives the short regulus profile of the spread corresponding to the translation plane group is the stabiliser of the spread in PGammaL(4,7) (the order of the group of the plane is this number times 6 times 49^2) rank the 7-rank of the projective plane paired whether it forms a polar pair or not (that is, if sigma is a polarity is S^sigma a different spread or not). Such a polar pair shares regulus profiles so are difficult to tell apart. the second line contains the spread in terms of LINE NUMBERS, ranging from 0 to 2849. The file linespg37 gives these lines in terms of point numbers, while the file linmatpg37 gives the 2 x 2 matrices in GL(2,7) corresponding to these numbers. The file linespg37 ------------------ Contains 2850 lines, each of length 8, of the form 0 1 2 5 6 3 4 7 0 8 9 12 13 10 11 14 0 15 16 19 20 17 18 21 This gives the point numbers corresponding to the lines. Therefore line 0 contains points {0,1,2,3,4,5,6,7}, line 1 contains points {0,8,9,10,11,12,13,14} and so on. The correspondence between point numbers and vectors in GF(7)^4 is simply lexicographic (0,0,0,1) -> 0 (0,0,1,0) -> 1 (0,0,1,1) -> 2 (0,0,1,2) -> 3 and so on. The file linmatpg37 ------------------- It is common to represent the spread by means of a "spread set". In this representation a projective line is represented as a 2-dimensional subspace of GF(7)^4. We can determine two special lines l_0 = {(0,0,x,y) | x,y in GF(7)} l_infinity = {(x,y,0,0) | x,y in GF(7)} while all the other lines are represented by l_A = {(x,xA) | x in GF(7)^2} In our representation, l_0 is the line 449 l_infinity is 0 and the file linmatpg37 contains the remaining mappings. Lines of the following form mean that 0 1 1 0 793 0 1 1 1 842 0 1 1 2 891 the matrix (0 1) (1 0) is the line 793 (0 1) (1 1) is the line 842 (0 1) (1 2) is the line 891 the line a b c d n (a b) (c d) is the line n