MinT - Best Known (237−212, 237, s)-Nets in Base 3

Best Known (237−212, 237, s)-Nets in Base 3

(237−212, 237, 32)-Net over F₃ — Constructive and digital

Digital (25, 237, 32)-net over F₃, using

t-expansion [i] based on digital (21, 237, 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]

(237−212, 237, 36)-Net over F₃ — Digital

Digital (25, 237, 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]

(237−212, 237, 61)-Net in Base 3 — Upper bound on s

There is no (25, 237, 62)-net in base 3, because

56 times m-reduction [i] would yield (25, 181, 62)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁸¹, 62, S₃, 3, 156), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 47306 133136 483029 642698 920914 502263 847013 801670 887417 165604 972327 618382 290023 516610 172221 / 157 > 3¹⁸¹ [i]