MinT - Best Known (68−42, 68, s)-Nets in Base 64

Best Known (68−42, 68, s)-Nets in Base 64

(68−42, 68, 193)-Net over F₆₄ — Constructive and digital

Digital (26, 68, 193)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 21, 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 (5, 47, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(68−42, 68, 288)-Net in Base 64 — Constructive

(26, 68, 288)-net in base 64, using

t-expansion [i] based on (22, 68, 288)-net in base 64, using
- 23 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]

(68−42, 68, 425)-Net over F₆₄ — Digital

Digital (26, 68, 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]

(68−42, 68, 513)-Net in Base 64

(26, 68, 513)-net in base 64, using

4 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]

(68−42, 68, 97205)-Net in Base 64 — Upper bound on s

There is no (26, 68, 97206)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 661 198077 102332 980283 777841 600842 450547 141800 516877 111889 220951 941993 117639 374075 902269 216592 771866 979945 149849 231156 571041 > 64⁶⁸ [i]