MinT - Best Known (42, 42+37, s)-Nets in Base 5

Best Known (42, 42+37, s)-Nets in Base 5

(42, 42+37, 78)-Net over F₅ — Constructive and digital

Digital (42, 79, 78)-net over F₅, using

t-expansion [i] based on digital (38, 79, 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+37, 108)-Net over F₅ — Digital

Digital (42, 79, 108)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+37, 2005)-Net in Base 5 — Upper bound on s

There is no (42, 79, 2006)-net in base 5, because

1 times m-reduction [i] would yield (42, 78, 2006)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3 334843 234316 430475 557495 176299 349132 687915 664963 978625 > 5⁷⁸ [i]