MinT - Best Known (67−47, 67, s)-Nets in Base 5

Best Known (67−47, 67, s)-Nets in Base 5

(67−47, 67, 43)-Net over F₅ — Constructive and digital

Digital (20, 67, 43)-net over F₅, using

t-expansion [i] based on digital (18, 67, 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]

(67−47, 67, 45)-Net over F₅ — Digital

Digital (20, 67, 45)-net over F₅, using

t-expansion [i] based on digital (19, 67, 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]

(67−47, 67, 213)-Net in Base 5 — Upper bound on s

There is no (20, 67, 214)-net in base 5, because

extracting embedded orthogonal array [i] would yield OA(5⁶⁷, 214, S₅, 47), but
- the linear programming bound shows that MÂ â‰¥ 4 882791 768041 267930 805220 059950 467467 556449 204308 973867 394853 831742 784646 689239 735906 085115 857422 351837 158203 125000 / 68 838348 844753 167738 422461 139858 320712 817410 014783 249069 295767 920377 > 5⁶⁷ [i]