MinT - Best Known (64, 64+43, s)-Nets in Base 25

Best Known (64, 64+43, s)-Nets in Base 25

(64, 64+43, 356)-Net over F₂₅ — Constructive and digital

Digital (64, 107, 356)-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 (10, 31, 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]
3. digital (10, 53, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(64, 64+43, 2527)-Net over F₂₅ — Digital

Digital (64, 107, 2527)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 64+43, 4116695)-Net in Base 25 — Upper bound on s

There is no (64, 107, 4116696)-net in base 25, because

1 times m-reduction [i] would yield (64, 106, 4116696)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 15192 911323 535682 956938 837051 669884 905147 668529 527831 424988 749563 452485 949806 457345 910466 182509 300541 677609 226735 326339 987507 310815 606922 572248 764225 > 25¹⁰⁶ [i]