MinT - Best Known (17, 17+23, s)-Nets in Base 64

Best Known (17, 17+23, s)-Nets in Base 64

Digital (17, 40, 208)-net over F₆₄, using

(17, 40, 288)-net in base 64, using

16 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 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 (17, 40, 295)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64⁴⁰, 295, F₆₄, 23) (dual of [295, 255, 24]-code), using
- 33 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 7 times 0, 1, 23 times 0) [i] based on linear OA(64³⁵, 257, F₆₄, 23) (dual of [257, 222, 24]-code), using
  - extended algebraic-geometric code AG_e(F,233P) [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]

(17, 40, 321)-net in base 64, using

20 times m-reduction [i] based on (17, 60, 321)-net in base 64, using
- base change [i] based on digital (2, 45, 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 (17, 40, 197421)-net in base 64, because

1 times m-reduction [i] would yield (17, 39, 197421)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 27608 471383 148612 797821 919349 584706 632879 457567 598604 381218 053106 163204 > 64³⁹ [i]