MinT - Best Known (63, 63+43, s)-Nets in Base 25

Best Known (63, 63+43, s)-Nets in Base 25

(63, 63+43, 334)-Net over F₂₅ — Constructive and digital

Digital (63, 106, 334)-net over F₂₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (9, 23, 104)-net over F₂₅, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by GarcÃa and Quoos [i]
2. digital (9, 30, 104)-net over F₂₅, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₂₅ (see above)
3. digital (10, 53, 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, 63+43, 2342)-Net over F₂₅ — Digital

Digital (63, 106, 2342)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(63, 63+43, 3531669)-Net in Base 25 — Upper bound on s

There is no (63, 106, 3531670)-net in base 25, because

1 times m-reduction [i] would yield (63, 105, 3531670)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 607 718823 969972 606087 539422 122943 557323 640393 314636 314927 199162 868107 080500 116109 125320 180839 458881 560100 808446 995163 065222 555339 930662 441872 123345 > 25¹⁰⁵ [i]