MinT - Best Known (49, 49+93, s)-Nets in Base 5

Best Known (49, 49+93, s)-Nets in Base 5

(49, 49+93, 82)-Net over F₅ — Constructive and digital

Digital (49, 142, 82)-net over F₅, using

t-expansion [i] based on digital (48, 142, 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+93, 96)-Net over F₅ — Digital

Digital (49, 142, 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+93, 591)-Net in Base 5 — Upper bound on s

There is no (49, 142, 592)-net in base 5, because

1 times m-reduction [i] would yield (49, 141, 592)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 376 832998 046366 257895 870070 035826 184171 339192 294180 067903 336742 883240 925131 639183 835296 623934 066945 > 5¹⁴¹ [i]