MinT - Best Known (56−41, 56, s)-Nets in Base 25

Best Known (56−41, 56, s)-Nets in Base 25

(56−41, 56, 126)-Net over F₂₅ — Constructive and digital

Digital (15, 56, 126)-net over F₂₅, using

t-expansion [i] based on digital (10, 56, 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]

(56−41, 56, 140)-Net over F₂₅ — Digital

Digital (15, 56, 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]

(56−41, 56, 2407)-Net in Base 25 — Upper bound on s

There is no (15, 56, 2408)-net in base 25, because

1 times m-reduction [i] would yield (15, 55, 2408)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 77336 863028 496880 992207 412848 915357 676586 833526 423803 742496 711122 738797 611265 > 25⁵⁵ [i]