MinT - Best Known (54, 54+89, s)-Nets in Base 5

Best Known (54, 54+89, s)-Nets in Base 5

(54, 54+89, 82)-Net over F₅ — Constructive and digital

Digital (54, 143, 82)-net over F₅, using

t-expansion [i] based on digital (48, 143, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(54, 54+89, 104)-Net over F₅ — Digital

Digital (54, 143, 104)-net over F₅, using

t-expansion [i] based on digital (51, 143, 104)-net over F₅, using
- net from sequence [i] based on digital (51, 103)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 51 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(54, 54+89, 745)-Net in Base 5 — Upper bound on s

There is no (54, 143, 746)-net in base 5, because

1 times m-reduction [i] would yield (54, 142, 746)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1861 178214 830932 445221 797834 468067 377506 485956 522651 451951 449260 247757 411658 210289 987147 141747 053505 > 5¹⁴² [i]