The nonexistance of linear codes with parametres 204, 4, 162 over GF (5)
The nonexistance of linear codes with parametres 204, 4, 162 over GF (5)

Summary/Abstract: The main problem of coding theory for four dimensional codes over the field with five elements is solved for all but three values of d: d=81,161,162. In this talk we announce the nonexistence of linear codes over GF(5) with parameters [204,4,162]. This problem is tackled from its geometric side. The existence of a code with these parameters is equivalent to that of a (204,42)-arc in PG(3,5). In order to rule out the existence of such arcs we consider a special dual arc which exhibits very strong divisibility properties. We prove that such an arc must have a hyperplane without 0-points which in turn implies the extendability of every (204,42)-arc to an arc with parameters (205,42). The nonexistence of the latter rules out the existence of (204,42)-arcs in PG(3,5). This result implies the exact value n5(4,162)=205, where nq(k,d) denotes the shortest length of a linear code of fixed dimension k and fixed minimum distance d over the field with q elements.

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