MinT - Best Known (78−25, 78, s)-Nets in Base 5

Best Known (78−25, 78, s)-Nets in Base 5

(78−25, 78, 252)-Net over F₅ — Constructive and digital

Digital (53, 78, 252)-net over F₅, using

8 times m-reduction [i] based on digital (53, 86, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 43, 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]

(78−25, 78, 470)-Net over F₅ — Digital

Digital (53, 78, 470)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−25, 78, 40389)-Net in Base 5 — Upper bound on s

There is no (53, 78, 40390)-net in base 5, because

1 times m-reduction [i] would yield (53, 77, 40390)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 661859 378509 673078 398791 081554 301605 678712 813628 731841 > 5⁷⁷ [i]