MinT - Best Known (89−70, 89, s)-Nets in Base 5

Best Known (89−70, 89, s)-Nets in Base 5

(89−70, 89, 43)-Net over F₅ — Constructive and digital

Digital (19, 89, 43)-net over F₅, using

t-expansion [i] based on digital (18, 89, 43)-net over F₅, using
- net from sequence [i] based on digital (18, 42)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₅ with g(F)Â =Â 17, N(F)Â =Â 42, and 1 place with degree 2 [i] based on function field F/F₅ with g(F) = 17 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(89−70, 89, 45)-Net over F₅ — Digital

Digital (19, 89, 45)-net over F₅, using

net from sequence [i] based on digital (19, 44)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 19 and N(F) ≥ 45, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(89−70, 89, 102)-Net in Base 5 — Upper bound on s

There is no (19, 89, 103)-net in base 5, because

1 times m-reduction [i] would yield (19, 88, 103)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁸⁸, 103, S₅, 69), but
  - the linear programming bound shows that MÂ â‰¥ 55 800890 235262 458139 981804 969157 255791 323279 936477 774754 166603 088378 906250 / 1 472522 247119 > 5⁸⁸ [i]