MinT - Best Known (50, 50+59, s)-Nets in Base 5

Best Known (50, 50+59, s)-Nets in Base 5

(50, 50+59, 82)-Net over F₅ — Constructive and digital

Digital (50, 109, 82)-net over F₅, using

t-expansion [i] based on digital (48, 109, 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]

(50, 50+59, 96)-Net over F₅ — Digital

Digital (50, 109, 96)-net over F₅, using

t-expansion [i] based on digital (49, 109, 96)-net over F₅, using
- net from sequence [i] based on digital (49, 95)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 49 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(50, 50+59, 1148)-Net in Base 5 — Upper bound on s

There is no (50, 109, 1149)-net in base 5, because

1 times m-reduction [i] would yield (50, 108, 1149)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3103 173542 825304 186806 837982 972044 899674 074258 409976 589527 390553 615764 441925 > 5¹⁰⁸ [i]