MinT - Best Known (38, 38+133, s)-Nets in Base 3

Best Known (38, 38+133, s)-Nets in Base 3

(38, 38+133, 38)-Net over F₃ — Constructive and digital

Digital (38, 171, 38)-net over F₃, using

t-expansion [i] based on digital (32, 171, 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+133, 52)-Net over F₃ — Digital

Digital (38, 171, 52)-net over F₃, using

t-expansion [i] based on digital (37, 171, 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+133, 122)-Net over F₃ — Upper bound on s (digital)

There is no digital (38, 171, 123)-net over F₃, because

52 times m-reduction [i] would yield digital (38, 119, 123)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹¹⁹, 123, F₃, 81) (dual of [123, 4, 82]-code), but
  - Griesmer bound [i]

(38, 38+133, 123)-Net in Base 3 — Upper bound on s

There is no (38, 171, 124)-net in base 3, because

63 times m-reduction [i] would yield (38, 108, 124)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁰⁸, 124, S₃, 70), but
  - the linear programming bound shows that MÂ â‰¥ 37 454952 569242 421982 996691 395370 215650 157240 469539 761140 263717 / 10878 767680 > 3¹⁰⁸ [i]