MinT - Best Known (78−50, 78, s)-Nets in Base 3

Best Known (78−50, 78, s)-Nets in Base 3

(78−50, 78, 37)-Net over F₃ — Constructive and digital

Digital (28, 78, 37)-net over F₃, using

t-expansion [i] based on digital (27, 78, 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]

(78−50, 78, 39)-Net over F₃ — Digital

Digital (28, 78, 39)-net over F₃, using

t-expansion [i] based on digital (27, 78, 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]

(78−50, 78, 99)-Net in Base 3 — Upper bound on s

There is no (28, 78, 100)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁷⁸, 100, S₃, 50), but
- the linear programming bound shows that MÂ â‰¥ 38101 440069 622014 833427 457285 684571 162543 165529 / 2181 771007 > 3⁷⁸ [i]