MinT - Best Known (88−65, 88, s)-Nets in Base 64

Best Known (88−65, 88, s)-Nets in Base 64

(88−65, 88, 177)-Net over F₆₄ — Constructive and digital

Digital (23, 88, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 88, 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]

(88−65, 88, 288)-Net in Base 64 — Constructive

(23, 88, 288)-net in base 64, using

t-expansion [i] based on (22, 88, 288)-net in base 64, using
- 3 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]

(88−65, 88, 342)-Net over F₆₄ — Digital

Digital (23, 88, 342)-net over F₆₄, using

t-expansion [i] based on digital (20, 88, 342)-net over F₆₄, using
- net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
    - a curve from the manYPoints database [i]

(88−65, 88, 16507)-Net in Base 64 — Upper bound on s

There is no (23, 88, 16508)-net in base 64, because

1 times m-reduction [i] would yield (23, 87, 16508)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 13 753584 364778 154989 351299 371075 310631 315262 917210 829078 699367 825736 198701 509776 724337 334004 182926 254963 568315 071908 389231 028837 397041 129453 166670 588953 024589 > 64⁸⁷ [i]