MinT - Best Known (84−69, 84, s)-Nets in Base 25

Best Known (84−69, 84, s)-Nets in Base 25

(84−69, 84, 126)-Net over F₂₅ — Constructive and digital

Digital (15, 84, 126)-net over F₂₅, using

t-expansion [i] based on digital (10, 84, 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]

(84−69, 84, 140)-Net over F₂₅ — Digital

Digital (15, 84, 140)-net over F₂₅, using

net from sequence [i] based on digital (15, 139)-sequence over F₂₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 15 and N(F) ≥ 140, using
  - an algebraic function field by Teo [i]

(84−69, 84, 1440)-Net in Base 25 — Upper bound on s

There is no (15, 84, 1441)-net in base 25, because

1 times m-reduction [i] would yield (15, 83, 1441)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 108 032055 213807 573627 931025 971259 803848 499875 912735 597537 893870 966170 795934 322745 322099 843842 150202 429557 507494 140785 > 25⁸³ [i]