MinT - Best Known (98−78, 98, s)-Nets in Base 5

Best Known (98−78, 98, s)-Nets in Base 5

(98−78, 98, 43)-Net over F₅ — Constructive and digital

Digital (20, 98, 43)-net over F₅, using

t-expansion [i] based on digital (18, 98, 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]

(98−78, 98, 45)-Net over F₅ — Digital

Digital (20, 98, 45)-net over F₅, using

t-expansion [i] based on digital (19, 98, 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]

(98−78, 98, 106)-Net in Base 5 — Upper bound on s

There is no (20, 98, 107)-net in base 5, because

3 times m-reduction [i] would yield (20, 95, 107)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁹⁵, 107, S₅, 75), but
  - the linear programming bound shows that MÂ â‰¥ 3749 470082 011763 089334 443281 903258 039788 884303 789018 304087 221622 467041 015625 / 1344 020328 > 5⁹⁵ [i]