MinT - Best Known (65−48, 65, s)-Nets in Base 64

Best Known (65−48, 65, s)-Nets in Base 64

(65−48, 65, 177)-Net over F₆₄ — Constructive and digital

Digital (17, 65, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 65, 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]

(65−48, 65, 257)-Net in Base 64 — Constructive

(17, 65, 257)-net in base 64, using

3 times m-reduction [i] based on (17, 68, 257)-net in base 64, using
- base change [i] based on digital (0, 51, 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]

(65−48, 65, 267)-Net over F₆₄ — Digital

Digital (17, 65, 267)-net over F₆₄, using

t-expansion [i] based on digital (16, 65, 267)-net over F₆₄, using
- net from sequence [i] based on digital (16, 266)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 16 and N(F) ≥ 267, using
    - a curve from the manYPoints database [i]

(65−48, 65, 12115)-Net in Base 64 — Upper bound on s

There is no (17, 65, 12116)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 2525 246105 768433 439699 195358 030836 505950 195182 161417 483648 342128 587511 346352 813379 727508 021616 460791 382627 904350 402344 > 64⁶⁵ [i]