MinT - Best Known (42, 42+99, s)-Nets in Base 5

Best Known (42, 42+99, s)-Nets in Base 5

(42, 42+99, 78)-Net over F₅ — Constructive and digital

Digital (42, 141, 78)-net over F₅, using

t-expansion [i] based on digital (38, 141, 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]

(42, 42+99, 80)-Net over F₅ — Digital

Digital (42, 141, 80)-net over F₅, using

t-expansion [i] based on digital (41, 141, 80)-net over F₅, using
- net from sequence [i] based on digital (41, 79)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 41 and N(F) ≥ 80, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(42, 42+99, 439)-Net in Base 5 — Upper bound on s

There is no (42, 141, 440)-net in base 5, because

1 times m-reduction [i] would yield (42, 140, 440)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 77 330828 667160 093879 861150 711924 778030 727187 402041 140159 221697 038928 533543 080189 923576 291692 884705 > 5¹⁴⁰ [i]