MinT - Best Known (78−39, 78, s)-Nets in Base 25

Best Known (78−39, 78, s)-Nets in Base 25

(78−39, 78, 252)-Net over F₂₅ — Constructive and digital

Digital (39, 78, 252)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (10, 29, 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, 49, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(78−39, 78, 479)-Net over F₂₅ — Digital

Digital (39, 78, 479)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−39, 78, 152866)-Net in Base 25 — Upper bound on s

There is no (39, 78, 152867)-net in base 25, because

1 times m-reduction [i] would yield (39, 77, 152867)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 437933 936364 559071 905864 863007 158596 688706 792702 044282 038302 356896 959414 513108 459642 068328 682065 387448 580825 > 25⁷⁷ [i]