MinT - Best Known (80−52, 80, s)-Nets in Base 3

Best Known (80−52, 80, s)-Nets in Base 3

(80−52, 80, 37)-Net over F₃ — Constructive and digital

Digital (28, 80, 37)-net over F₃, using

t-expansion [i] based on digital (27, 80, 37)-net over F₃, using
- net from sequence [i] based on digital (27, 36)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(80−52, 80, 39)-Net over F₃ — Digital

Digital (28, 80, 39)-net over F₃, using

t-expansion [i] based on digital (27, 80, 39)-net over F₃, using
- net from sequence [i] based on digital (27, 38)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 27 and N(F) ≥ 39, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(80−52, 80, 97)-Net in Base 3 — Upper bound on s

There is no (28, 80, 98)-net in base 3, because

1 times m-reduction [i] would yield (28, 79, 98)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁷⁹, 98, S₃, 51), but
  - the linear programming bound shows that MÂ â‰¥ 37359 886424 244488 451760 980773 909770 698911 495751 / 677 763515 > 3⁷⁹ [i]