MinT - Best Known (138−83, 138, s)-Nets in Base 5

Best Known (138−83, 138, s)-Nets in Base 5

(138−83, 138, 82)-Net over F₅ — Constructive and digital

Digital (55, 138, 82)-net over F₅, using

t-expansion [i] based on digital (48, 138, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(138−83, 138, 108)-Net over F₅ — Digital

Digital (55, 138, 108)-net over F₅, using

net from sequence [i] based on digital (55, 107)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 55 and N(F) ≥ 108, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(138−83, 138, 843)-Net in Base 5 — Upper bound on s

There is no (55, 138, 844)-net in base 5, because

1 times m-reduction [i] would yield (55, 137, 844)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 574931 068148 829908 072899 368725 347790 682370 809838 990100 567825 426704 613941 206750 038892 155122 383921 > 5¹³⁷ [i]