MinT - Best Known (86−66, 86, s)-Nets in Base 5

Best Known (86−66, 86, s)-Nets in Base 5

(86−66, 86, 43)-Net over F₅ — Constructive and digital

Digital (20, 86, 43)-net over F₅, using

t-expansion [i] based on digital (18, 86, 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]

(86−66, 86, 45)-Net over F₅ — Digital

Digital (20, 86, 45)-net over F₅, using

t-expansion [i] based on digital (19, 86, 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]

(86−66, 86, 119)-Net in Base 5 — Upper bound on s

There is no (20, 86, 120)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁸⁶, 120, S₅, 66), but
- the linear programming bound shows that MÂ â‰¥ 3 659188 660547 740119 702804 058000 671653 918454 140852 627750 994663 369255 022189 463488 757610 321044 921875 / 2 764137 695491 693048 028523 660647 380371 > 5⁸⁶ [i]