MinT - Best Known (5, ∞, s)-Nets in Base 7

Best Known (5, ∞, s)-Nets in Base 7

(5, ∞, 13)-Net over F₇ — Constructive and digital

Digital (5, m, 13)-net over F₇ for arbitrarily large m, using

net from sequence [i] based on digital (5, 12)-sequence over F₇, using
- Niederreiter sequence [i]

(5, ∞, 24)-Net over F₇ — Digital

Digital (5, m, 24)-net over F₇ for arbitrarily large m, using

net from sequence [i] based on digital (5, 23)-sequence over F₇, using
- t-expansion [i] based on digital (4, 23)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 4 and N(F) ≥ 24, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(5, ∞, 44)-Net in Base 7 — Upper bound on s

There is no (5, m, 45)-net in base 7 for arbitrarily large m, because

m-reduction [i] would yield (5, 86, 45)-net in base 7, but
- extracting embedded OOA [i] would yield OOA(7⁸⁶, 45, S₇, 2, 81), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 405 368844 447017 397144 485951 958074 725759 929366 921792 254455 742666 234993 920165 / 82 > 7⁸⁶ [i]