MinT - Best Known (126−83, 126, s)-Nets in Base 5

Best Known (126−83, 126, s)-Nets in Base 5

(126−83, 126, 78)-Net over F₅ — Constructive and digital

Digital (43, 126, 78)-net over F₅, using

t-expansion [i] based on digital (38, 126, 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]

(126−83, 126, 84)-Net over F₅ — Digital

Digital (43, 126, 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]

(126−83, 126, 515)-Net in Base 5 — Upper bound on s

There is no (43, 126, 516)-net in base 5, because

1 times m-reduction [i] would yield (43, 125, 516)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2366 470373 404914 384146 094359 847210 086090 816170 118032 599086 724923 434853 845529 572123 902225 > 5¹²⁵ [i]