MinT - Best Known (81−41, 81, s)-Nets in Base 25

Best Known (81−41, 81, s)-Nets in Base 25

(81−41, 81, 252)-Net over F₂₅ — Constructive and digital

Digital (40, 81, 252)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (10, 30, 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, 51, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(81−41, 81, 456)-Net over F₂₅ — Digital

Digital (40, 81, 456)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(81−41, 81, 135151)-Net in Base 25 — Upper bound on s

There is no (40, 81, 135152)-net in base 25, because

1 times m-reduction [i] would yield (40, 80, 135152)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 6842 484386 074130 932808 916369 432029 398697 666022 237992 134050 011485 633585 697672 929962 887535 245174 872670 937215 829505 > 25⁸⁰ [i]