MinT - Best Known (38, 38+69, s)-Nets in Base 3

Best Known (38, 38+69, s)-Nets in Base 3

(38, 38+69, 38)-Net over F₃ — Constructive and digital

Digital (38, 107, 38)-net over F₃, using

t-expansion [i] based on digital (32, 107, 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]

(38, 38+69, 52)-Net over F₃ — Digital

Digital (38, 107, 52)-net over F₃, using

t-expansion [i] based on digital (37, 107, 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]

(38, 38+69, 125)-Net in Base 3 — Upper bound on s

There is no (38, 107, 126)-net in base 3, because

1 times m-reduction [i] would yield (38, 106, 126)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁰⁶, 126, S₃, 68), but
  - the linear programming bound shows that MÂ â‰¥ 1 061877 790063 016487 101437 522958 614020 021078 388104 588955 021944 / 2475 548075 > 3¹⁰⁶ [i]