MinT - Best Known (41, 41+41, s)-Nets in Base 5

Best Known (41, 41+41, s)-Nets in Base 5

(41, 41+41, 78)-Net over F₅ — Constructive and digital

Digital (41, 82, 78)-net over F₅, using

t-expansion [i] based on digital (38, 82, 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]

(41, 41+41, 89)-Net over F₅ — Digital

Digital (41, 82, 89)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(41, 41+41, 1391)-Net in Base 5 — Upper bound on s

There is no (41, 82, 1392)-net in base 5, because

1 times m-reduction [i] would yield (41, 81, 1392)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 414 512736 808412 417354 019194 764604 460513 801962 764359 092225 > 5⁸¹ [i]