MinT - Best Known (26−15, 26, s)-Nets in Base 64

Best Known (26−15, 26, s)-Nets in Base 64

(26−15, 26, 184)-Net over F₆₄ — Constructive and digital

Digital (11, 26, 184)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]
2. digital (3, 18, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(26−15, 26, 236)-Net over F₆₄ — Digital

Digital (11, 26, 236)-net over F₆₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(64²⁶, 236, F₆₄, 15) (dual of [236, 210, 16]-code), using
- 10 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 9 times 0) [i] based on linear OA(64²⁵, 225, F₆₄, 15) (dual of [225, 200, 16]-code), using
  - extended algebraic-geometric code AG_e(F,209P) [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(26−15, 26, 261)-Net in Base 64 — Constructive

(11, 26, 261)-net in base 64, using

2 times m-reduction [i] based on (11, 28, 261)-net in base 64, using
- base change [i] based on digital (4, 21, 261)-net over F₂₅₆, using
  - net from sequence [i] based on digital (4, 260)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(26−15, 26, 321)-Net in Base 64

(11, 26, 321)-net in base 64, using

10 times m-reduction [i] based on (11, 36, 321)-net in base 64, using
- base change [i] based on digital (2, 27, 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]

(26−15, 26, 151430)-Net in Base 64 — Upper bound on s

There is no (11, 26, 151431)-net in base 64, because

1 times m-reduction [i] would yield (11, 25, 151431)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1427 313252 238008 716094 211145 733296 198340 642112 > 64²⁵ [i]