MinT - Best Known (137−116, 137, s)-Nets in Base 5

Best Known (137−116, 137, s)-Nets in Base 5

(137−116, 137, 43)-Net over F₅ — Constructive and digital

Digital (21, 137, 43)-net over F₅, using

t-expansion [i] based on digital (18, 137, 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]

(137−116, 137, 50)-Net over F₅ — Digital

Digital (21, 137, 50)-net over F₅, using

net from sequence [i] based on digital (21, 49)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 21 and N(F) ≥ 50, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(137−116, 137, 111)-Net in Base 5 — Upper bound on s

There is no (21, 137, 112)-net in base 5, because

40 times m-reduction [i] would yield (21, 97, 112)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁹⁷, 112, S₅, 76), but
  - the linear programming bound shows that MÂ â‰¥ 29 282971 382900 591824 936232 851882 769051 034272 056455 165511 579252 779483 795166 015625 / 409917 357801 > 5⁹⁷ [i]