MinT - Best Known (172−144, 172, s)-Nets in Base 3

Best Known (172−144, 172, s)-Nets in Base 3

(172−144, 172, 37)-Net over F₃ — Constructive and digital

Digital (28, 172, 37)-net over F₃, using

t-expansion [i] based on digital (27, 172, 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]

(172−144, 172, 39)-Net over F₃ — Digital

Digital (28, 172, 39)-net over F₃, using

t-expansion [i] based on digital (27, 172, 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]

(172−144, 172, 72)-Net in Base 3 — Upper bound on s

There is no (28, 172, 73)-net in base 3, because

30 times m-reduction [i] would yield (28, 142, 73)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁴², 73, S₃, 2, 114), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 6597 874218 733402 809356 804076 642052 639813 649252 491630 534427 156747 493453 / 115 > 3¹⁴² [i]