MinT - Best Known (126−65, 126, s)-Nets in Base 5

Best Known (126−65, 126, s)-Nets in Base 5

(126−65, 126, 82)-Net over F₅ — Constructive and digital

Digital (61, 126, 82)-net over F₅, using

t-expansion [i] based on digital (48, 126, 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]

(126−65, 126, 120)-Net over F₅ — Digital

Digital (61, 126, 120)-net over F₅, using

net from sequence [i] based on digital (61, 119)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 61 and N(F) ≥ 120, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(126−65, 126, 1695)-Net in Base 5 — Upper bound on s

There is no (61, 126, 1696)-net in base 5, because

1 times m-reduction [i] would yield (61, 125, 1696)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2386 135047 362899 076249 461213 383188 029908 588880 055504 711311 916727 716035 359710 788751 704065 > 5¹²⁵ [i]