MinT - Best Known (83−68, 83, s)-Nets in Base 5

Best Known (83−68, 83, s)-Nets in Base 5

(83−68, 83, 36)-Net over F₅ — Constructive and digital

Digital (15, 83, 36)-net over F₅, using

net from sequence [i] based on digital (15, 35)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₅ with g(F)Â =Â 11, N(F)Â =Â 32, and 4 places with degree 2 [i] based on function field F/F₅ with g(F) = 11 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(83−68, 83, 39)-Net over F₅ — Digital

Digital (15, 83, 39)-net over F₅, using

t-expansion [i] based on digital (14, 83, 39)-net over F₅, using
- net from sequence [i] based on digital (14, 38)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 14 and N(F) ≥ 39, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(83−68, 83, 83)-Net in Base 5 — Upper bound on s

There is no (15, 83, 84)-net in base 5, because

11 times m-reduction [i] would yield (15, 72, 84)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁷², 84, S₅, 57), but
  - the linear programming bound shows that MÂ â‰¥ 1 900530 175401 836385 784548 610899 946652 352809 906005 859375 / 8671 > 5⁷² [i]