MinT - Best Known (112−69, 112, s)-Nets in Base 5

Best Known (112−69, 112, s)-Nets in Base 5

(112−69, 112, 78)-Net over F₅ — Constructive and digital

Digital (43, 112, 78)-net over F₅, using

t-expansion [i] based on digital (38, 112, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(112−69, 112, 84)-Net over F₅ — Digital

Digital (43, 112, 84)-net over F₅, using

net from sequence [i] based on digital (43, 83)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 43 and N(F) ≥ 84, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(112−69, 112, 622)-Net in Base 5 — Upper bound on s

There is no (43, 112, 623)-net in base 5, because

1 times m-reduction [i] would yield (43, 111, 623)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 385220 259416 262365 974899 447299 728179 614356 165811 652673 081650 071126 816093 091705 > 5¹¹¹ [i]