MinT - Best Known (15, 15+19, s)-Nets in Base 64

Best Known (15, 15+19, s)-Nets in Base 64

Digital (15, 34, 208)-net over F₆₄, using

(15, 34, 288)-net in base 64, using

8 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
- base change [i] based on digital (9, 36, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

Digital (15, 34, 320)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³⁴, 320, F₆₄, 19) (dual of [320, 286, 20]-code), using
- 60 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 12 times 0, 1, 46 times 0) [i] based on linear OA(64³¹, 257, F₆₄, 19) (dual of [257, 226, 20]-code), using
  - extended algebraic-geometric code AG_e(F,237P) [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(15, 34, 321)-net in base 64, using

18 times m-reduction [i] based on (15, 52, 321)-net in base 64, using
- base change [i] based on digital (2, 39, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

There is no (15, 34, 276099)-net in base 64, because

1 times m-reduction [i] would yield (15, 33, 276099)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 401739 064408 020279 590060 484508 382947 376730 377615 382451 064224 > 64³³ [i]