MinT - Best Known (65−51, 65, s)-Nets in Base 81

Best Known (65−51, 65, s)-Nets in Base 81

(65−51, 65, 224)-Net over F₈₁ — Constructive and digital

Digital (14, 65, 224)-net over F₈₁, using

t-expansion [i] based on digital (13, 65, 224)-net over F₈₁, using
- net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
    - a function field by SÃ©mirat [i]

(65−51, 65, 298)-Net over F₈₁ — Digital

Digital (14, 65, 298)-net over F₈₁, using

t-expansion [i] based on digital (12, 65, 298)-net over F₈₁, using
- net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
    - a curve from the manYPoints database [i]

(65−51, 65, 9765)-Net in Base 81 — Upper bound on s

There is no (14, 65, 9766)-net in base 81, because

1 times m-reduction [i] would yield (14, 64, 9766)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 139 058783 662559 823727 983168 411974 189866 231780 795823 722207 078480 422386 568574 799358 428191 330989 821201 997851 179801 529762 252001 > 81⁶⁴ [i]