MinT - Best Known (68−62, 68, s)-Nets in Base 64

Best Known (68−62, 68, s)-Nets in Base 64

(68−62, 68, 128)-Net over F₆₄ — Constructive and digital

Digital (6, 68, 128)-net over F₆₄, using

t-expansion [i] based on digital (5, 68, 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]

(68−62, 68, 161)-Net over F₆₄ — Digital

Digital (6, 68, 161)-net over F₆₄, using

net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(68−62, 68, 1776)-Net in Base 64 — Upper bound on s

There is no (6, 68, 1777)-net in base 64, because

6 times m-reduction [i] would yield (6, 62, 1777)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 9628 243677 332979 985102 055123 697968 863615 149142 257563 526399 934787 098909 501690 444032 902233 177985 774325 020091 801976 > 64⁶² [i]