MinT - Best Known (85−65, 85, s)-Nets in Base 5

Best Known (85−65, 85, s)-Nets in Base 5

(85−65, 85, 43)-Net over F₅ — Constructive and digital

Digital (20, 85, 43)-net over F₅, using

t-expansion [i] based on digital (18, 85, 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]

(85−65, 85, 45)-Net over F₅ — Digital

Digital (20, 85, 45)-net over F₅, using

t-expansion [i] based on digital (19, 85, 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]

(85−65, 85, 125)-Net in Base 5 — Upper bound on s

There is no (20, 85, 126)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁸⁵, 126, S₅, 65), but
- the linear programming bound shows that MÂ â‰¥ 144 471513 340065 449263 037885 952010 651258 545437 840641 472341 144204 739882 273003 104273 811914 026737 213134 765625 / 470 101103 972684 112746 856024 521475 589350 417573 > 5⁸⁵ [i]