MinT - Best Known (49, 49+101, s)-Nets in Base 5

Best Known (49, 49+101, s)-Nets in Base 5

(49, 49+101, 82)-Net over F₅ — Constructive and digital

Digital (49, 150, 82)-net over F₅, using

t-expansion [i] based on digital (48, 150, 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]

(49, 49+101, 96)-Net over F₅ — Digital

Digital (49, 150, 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]

(49, 49+101, 553)-Net in Base 5 — Upper bound on s

There is no (49, 150, 554)-net in base 5, because

1 times m-reduction [i] would yield (49, 149, 554)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 147 870247 881824 156852 162179 046741 631460 076515 816578 833488 596716 972660 312114 541870 934687 581853 653728 089281 > 5¹⁴⁹ [i]