It is known that 42 is the largest size of a 6-arc in the Desarguesian projective plane of order 8. In this paper, we classify these (42, 6) 8 arcs. Equivalently, we classify the smallest 3-fold blocking sets in PG(2, 8), which have size 31.
We prove that every [n, k, d] q code with q ≥ 4, k ≥ 3, whose weights are congruent to 0, −1 or −2 modulo q and d ≡ −1 (mod q) is extendable unless its diversity is q 2 q k−3 + θ k−3 , q 2 q k−3 for odd q, where θ j = (q j+1 − 1)/(q − 1).
Hill and Kolev give a large class of q-ary linear codes meeting the Griesmer bound, which are called codes of Belov type (Hill and Kolev, Chapman Hall/CRC Research Notes in Mathematics 403, pp. 127-152, 1999). In this article, we prove that there are no linear codes meeting the Griesmer bound for values of d close to those for codes of Belov type. So we conclude that the lower bounds of d of codes of Belov type are sharp. We give a large class of length optimal codes with n q (k, d) = g q (k, d) + 1.
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