MinT - Best Known (15, 35, s)-Nets in Base 64

Best Known (15, 35, s)-Nets in Base 64

Digital (15, 35, 193)-net over F₆₄, using

Digital (15, 35, 285)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64³⁵, 285, F₆₄, 20) (dual of [285, 250, 21]-code), using
- 25 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 21 times 0) [i] based on linear OA(64³², 257, F₆₄, 20) (dual of [257, 225, 21]-code), using
  - extended algebraic-geometric code AG_e(F,236P) [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, 35, 288)-net in base 64, using

7 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]

(15, 35, 321)-net in base 64, using

17 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, 35, 150749)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1645 587848 124662 384640 561660 703037 789195 753453 991315 079828 847446 > 64³⁵ [i]