MinT - Best Known (163−138, 163, s)-Nets in Base 3

Best Known (163−138, 163, s)-Nets in Base 3

(163−138, 163, 32)-Net over F₃ — Constructive and digital

Digital (25, 163, 32)-net over F₃, using

t-expansion [i] based on digital (21, 163, 32)-net over F₃, using
- net from sequence [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(163−138, 163, 36)-Net over F₃ — Digital

Digital (25, 163, 36)-net over F₃, using

net from sequence [i] based on digital (25, 35)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 25 and N(F) ≥ 36, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(163−138, 163, 65)-Net in Base 3 — Upper bound on s

There is no (25, 163, 66)-net in base 3, because

35 times m-reduction [i] would yield (25, 128, 66)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹²⁸, 66, S₃, 2, 103), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 318 334983 598941 745631 063567 258138 003973 311712 991447 709945 617947 / 26 > 3¹²⁸ [i]