MinT - Best Known (21, 21+25, s)-Nets in Base 64

Best Known (21, 21+25, s)-Nets in Base 64

(21, 21+25, 257)-Net over F₆₄ — Constructive and digital

Digital (21, 46, 257)-net over F₆₄, using

1 times m-reduction [i] based on digital (21, 47, 257)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 14, 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 (7, 33, 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]

(21, 21+25, 322)-Net in Base 64 — Constructive

(21, 46, 322)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. digital (0, 12, 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. (9, 34, 257)-net in base 64, using
  - 2 times m-reduction [i] based on (9, 36, 257)-net in base 64, using
    - base change [i] based on digital (0, 27, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(21, 21+25, 463)-Net over F₆₄ — Digital

Digital (21, 46, 463)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(21, 21+25, 513)-Net in Base 64

(21, 46, 513)-net in base 64, using

6 times m-reduction [i] based on (21, 52, 513)-net in base 64, using
- base change [i] based on digital (8, 39, 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]

(21, 21+25, 497955)-Net in Base 64 — Upper bound on s

There is no (21, 46, 497956)-net in base 64, because

1 times m-reduction [i] would yield (21, 45, 497956)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1897 146288 515144 944278 775346 266826 926492 751261 373034 884272 029569 740224 730138 020120 > 64⁴⁵ [i]