MinT - Best Known (90−70, 90, s)-Nets in Base 5

Best Known (90−70, 90, s)-Nets in Base 5

(90−70, 90, 43)-Net over F₅ — Constructive and digital

Digital (20, 90, 43)-net over F₅, using

t-expansion [i] based on digital (18, 90, 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]

(90−70, 90, 45)-Net over F₅ — Digital

Digital (20, 90, 45)-net over F₅, using

t-expansion [i] based on digital (19, 90, 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]

(90−70, 90, 109)-Net in Base 5 — Upper bound on s

There is no (20, 90, 110)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁹⁰, 110, S₅, 70), but
- the linear programming bound shows that MÂ â‰¥ 296 592041 988785 376481 669670 541067 203588 421474 780290 054695 797152 817249 298095 703125 / 349849 859348 320149 > 5⁹⁰ [i]