MinT - Best Known (65−25, 65, s)-Nets in Base 25

Best Known (65−25, 65, s)-Nets in Base 25

(65−25, 65, 278)-Net over F₂₅ — Constructive and digital

Digital (40, 65, 278)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 8, 26)-net over F₂₅, using
  - net from sequence [i] based on digital (0, 25)-sequence over F₂₅, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 0 and N(F) ≥ 26, using
      - the rational function field F₂₅(x) [i]
    - Niederreiter sequence [i]
2. digital (10, 22, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]
3. digital (10, 35, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(65−25, 65, 2508)-Net over F₂₅ — Digital

Digital (40, 65, 2508)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(65−25, 65, 6292595)-Net in Base 25 — Upper bound on s

There is no (40, 65, 6292596)-net in base 25, because

1 times m-reduction [i] would yield (40, 64, 6292596)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 293873 959324 002170 699933 728003 826043 799023 898561 497842 907402 113905 316574 263708 013953 948545 > 25⁶⁴ [i]