MinT - Best Known (39, 178, s)-Nets in Base 3

Best Known (39, 178, s)-Nets in Base 3

(39, 178, 42)-Net over F₃ — Constructive and digital

Digital (39, 178, 42)-net over F₃, using

net from sequence [i] based on digital (39, 41)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
  - an explicitly constructive algebraic function field [i]

(39, 178, 52)-Net over F₃ — Digital

Digital (39, 178, 52)-net over F₃, using

t-expansion [i] based on digital (37, 178, 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]

(39, 178, 124)-Net over F₃ — Upper bound on s (digital)

There is no digital (39, 178, 125)-net over F₃, because

58 times m-reduction [i] would yield digital (39, 120, 125)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹²⁰, 125, F₃, 81) (dual of [125, 5, 82]-code), but
  - Griesmer bound [i]

(39, 178, 125)-Net in Base 3 — Upper bound on s

There is no (39, 178, 126)-net in base 3, because

64 times m-reduction [i] would yield (39, 114, 126)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹¹⁴, 126, S₃, 75), but
  - the linear programming bound shows that MÂ â‰¥ 30 712703 035837 543817 171656 466148 659734 050649 950249 413259 499491 / 12 402877 > 3¹¹⁴ [i]