MinT - Best Known (26, 26+39, s)-Nets in Base 64

Best Known (26, 26+39, s)-Nets in Base 64

(26, 26+39, 242)-Net over F₆₄ — Constructive and digital

Digital (26, 65, 242)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 46, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(26, 26+39, 288)-Net in Base 64 — Constructive

(26, 65, 288)-net in base 64, using

t-expansion [i] based on (22, 65, 288)-net in base 64, using
- 26 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
  - base change [i] based on digital (9, 78, 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]

(26, 26+39, 425)-Net over F₆₄ — Digital

Digital (26, 65, 425)-net over F₆₄, using

net from sequence [i] based on digital (26, 424)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 26 and N(F) ≥ 425, using
  - a curve from the manYPoints database [i]

(26, 26+39, 513)-Net in Base 64

(26, 65, 513)-net in base 64, using

7 times m-reduction [i] based on (26, 72, 513)-net in base 64, using
- base change [i] based on digital (8, 54, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(26, 26+39, 152694)-Net in Base 64 — Upper bound on s

There is no (26, 65, 152695)-net in base 64, because

1 times m-reduction [i] would yield (26, 64, 152695)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 39 404875 699280 865859 924505 925461 536737 362121 221634 869802 337180 493048 201401 594661 458949 769349 196737 311247 974660 258292 > 64⁶⁴ [i]