MinT - Best Known (85, 85+33, s)-Nets in Base 4

Best Known (85, 85+33, s)-Nets in Base 4

Digital (85, 118, 384)-net over F₄, using

2 times m-reduction [i] based on digital (85, 120, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 40, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(85, 118, 387)-net in base 4, using

Digital (85, 118, 724)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁸, 724, F₄, 33) (dual of [724, 606, 34]-code), using
- 605 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (9, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 23 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0, 1, 28 times 0) [i] based on linear OA(4³³, 34, F₄, 33) (dual of [34, 1, 34]-code or 34-arc in PG(32,4)), using
  - dual of repetition code with length 34 [i]

There is no (85, 118, 57265)-net in base 4, because

1 times m-reduction [i] would yield (85, 117, 57265)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 27612 494917 686127 578077 744148 147125 645397 885307 588854 493935 966554 203735 > 4¹¹⁷ [i]