MinT - Best Known (14, 34, s)-Nets in Base 64

Best Known (14, 34, s)-Nets in Base 64

Digital (14, 34, 184)-net over F₆₄, using

Digital (14, 34, 258)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(64³⁴, 258, F₆₄, 2, 20) (dual of [(258, 2), 482, 21]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(64³², 257, F₆₄, 2, 20) (dual of [(257, 2), 482, 21]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(2;F,493P) [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]

(14, 34, 288)-net in base 64, using

1 times m-reduction [i] based on (14, 35, 288)-net in base 64, using
- base change [i] based on digital (9, 30, 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]

(14, 34, 321)-net in base 64, using

14 times m-reduction [i] based on (14, 48, 321)-net in base 64, using
- base change [i] based on digital (2, 36, 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 (14, 34, 99456)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 25 713178 933231 245635 632797 553037 235507 134431 645614 474275 593105 > 64³⁴ [i]