MinT - Best Known (86−41, 86, s)-Nets in Base 5

Best Known (86−41, 86, s)-Nets in Base 5

(86−41, 86, 78)-Net over F₅ — Constructive and digital

Digital (45, 86, 78)-net over F₅, using

t-expansion [i] based on digital (38, 86, 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]

(86−41, 86, 109)-Net over F₅ — Digital

Digital (45, 86, 109)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−41, 86, 1925)-Net in Base 5 — Upper bound on s

There is no (45, 86, 1926)-net in base 5, because

1 times m-reduction [i] would yield (45, 85, 1926)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 258924 692663 526792 175915 491800 364656 063705 072854 612304 779585 > 5⁸⁵ [i]