MinT - Best Known (67−29, 67, s)-Nets in Base 25

Best Known (67−29, 67, s)-Nets in Base 25

(67−29, 67, 252)-Net over F₂₅ — Constructive and digital

Digital (38, 67, 252)-net over F₂₅, using

7 times m-reduction [i] based on digital (38, 74, 252)-net over F₂₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 28, 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]
  2. digital (10, 46, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(67−29, 67, 1056)-Net over F₂₅ — Digital

Digital (38, 67, 1056)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(67−29, 67, 980690)-Net in Base 25 — Upper bound on s

There is no (38, 67, 980691)-net in base 25, because

1 times m-reduction [i] would yield (38, 66, 980691)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 183 672274 076933 024939 224607 597245 559178 567289 342729 757920 046653 335640 013909 685431 143180 919665 > 25⁶⁶ [i]