MinT - Best Known (69−49, 69, s)-Nets in Base 5

Best Known (69−49, 69, s)-Nets in Base 5

(69−49, 69, 43)-Net over F₅ — Constructive and digital

Digital (20, 69, 43)-net over F₅, using

t-expansion [i] based on digital (18, 69, 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]

(69−49, 69, 45)-Net over F₅ — Digital

Digital (20, 69, 45)-net over F₅, using

t-expansion [i] based on digital (19, 69, 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]

(69−49, 69, 209)-Net in Base 5 — Upper bound on s

There is no (20, 69, 210)-net in base 5, because

1 times m-reduction [i] would yield (20, 68, 210)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁶⁸, 210, S₅, 48), but
  - the linear programming bound shows that MÂ â‰¥ 6760 956083 916702 352992 250564 109932 606189 415621 657408 476429 552982 124466 864951 923836 257741 129429 632565 006613 731384 277343 750000 / 19342 287027 559923 481797 783576 010130 384623 417931 322127 496147 355024 935112 515561 > 5⁶⁸ [i]