MinT - Best Known (111−57, 111, s)-Nets in Base 5

Best Known (111−57, 111, s)-Nets in Base 5

(111−57, 111, 82)-Net over F₅ — Constructive and digital

Digital (54, 111, 82)-net over F₅, using

t-expansion [i] based on digital (48, 111, 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]

(111−57, 111, 104)-Net over F₅ — Digital

Digital (54, 111, 104)-net over F₅, using

t-expansion [i] based on digital (51, 111, 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]

(111−57, 111, 1553)-Net in Base 5 — Upper bound on s

There is no (54, 111, 1554)-net in base 5, because

1 times m-reduction [i] would yield (54, 110, 1554)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 78126 569739 387264 350796 986848 480969 606670 618216 935899 692209 990761 310085 646017 > 5¹¹⁰ [i]