# New partial projective plane

In 2011, Roy completed a search for projective planes of order ten.  His search for weight 15 codewords was based on completing the last 36 columns of the representation given by MacWilliams, Sloane, and Thompson.  These columns may be split into six blocks of six columns each, where each block consists of the columns incident to one of the first six rows.  The entries in the first block may be uniquely specified up to equivalence by using row permutations to order the rows in lexicographic order.  Roy finds exactly 79 inequivalent ways of completing the second block.  His search attempts to extend each of the 79 cases to the later blocks, but in each case finds no way to perform the extension to the fourth block.

In order to verify his search we derived a SAT instance attempting to complete columns 76–111 and all rows 1–111.  Surprisingly, the SAT instance generated in this way was found to be satisfiable.  We found a completion of all six blocks, and therefore there should exist some symmetry of this completion that transforms the first two blocks into one of the 79 representatives used in the previous search (regardless of how those representatives were chosen).  Thus, we are forced to conclude that the previous search was incomplete.

The following is an explicit representation of an 111 by 51 partial projective plane that we found, where columns 76–111 have been ordered to appear immediately following columns 1–15.

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