MinT - Best Known (63−25, 63, s)-Nets in Base 25

Best Known (63−25, 63, s)-Nets in Base 25

(63−25, 63, 258)-Net over F₂₅ — Constructive and digital

Digital (38, 63, 258)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 12, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]
2. digital (4, 16, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅ (see above)
3. digital (10, 35, 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]

(63−25, 63, 1921)-Net over F₂₅ — Digital

Digital (38, 63, 1921)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(63−25, 63, 3679929)-Net in Base 25 — Upper bound on s

There is no (38, 63, 3679930)-net in base 25, because

1 times m-reduction [i] would yield (38, 62, 3679930)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 470 198375 543743 444437 968476 648929 917825 091672 732745 341133 716399 420992 526398 813988 529345 > 25⁶² [i]