MinT - Best Known (172−135, 172, s)-Nets in Base 3

Best Known (172−135, 172, s)-Nets in Base 3

(172−135, 172, 38)-Net over F₃ — Constructive and digital

Digital (37, 172, 38)-net over F₃, using

t-expansion [i] based on digital (32, 172, 38)-net over F₃, using
- net from sequence [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(172−135, 172, 52)-Net over F₃ — Digital

Digital (37, 172, 52)-net over F₃, using

net from sequence [i] based on digital (37, 51)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 37 and N(F) ≥ 52, using
  - a curve from the manYPoints database [i]

(172−135, 172, 120)-Net in Base 3 — Upper bound on s

There is no (37, 172, 121)-net in base 3, because

64 times m-reduction [i] would yield (37, 108, 121)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁰⁸, 121, S₃, 71), but
  - the linear programming bound shows that MÂ â‰¥ 16 463997 226942 154563 727248 349077 822257 696630 163050 643509 / 3649 > 3¹⁰⁸ [i]