MinT - Best Known (31−21, 31, s)-Nets in Base 5

Best Known (31−21, 31, s)-Nets in Base 5

(31−21, 31, 26)-Net over F₅ — Constructive and digital

Digital (10, 31, 26)-net over F₅, using

t-expansion [i] based on digital (9, 31, 26)-net over F₅, using
- net from sequence [i] based on digital (9, 25)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 9 and N(F) ≥ 26, using
    - an explicitly constructive algebraic function field [i]

(31−21, 31, 27)-Net over F₅ — Digital

Digital (10, 31, 27)-net over F₅, using

net from sequence [i] based on digital (10, 26)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 10 and N(F) ≥ 27, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(31−21, 31, 134)-Net in Base 5 — Upper bound on s

There is no (10, 31, 135)-net in base 5, because

1 times m-reduction [i] would yield (10, 30, 135)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 954 729073 434203 352025 > 5³⁰ [i]