MinT - Best Known (72−47, 72, s)-Nets in Base 3

Best Known (72−47, 72, s)-Nets in Base 3

(72−47, 72, 32)-Net over F₃ — Constructive and digital

Digital (25, 72, 32)-net over F₃, using

t-expansion [i] based on digital (21, 72, 32)-net over F₃, using
- net from sequence [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(72−47, 72, 36)-Net over F₃ — Digital

Digital (25, 72, 36)-net over F₃, using

net from sequence [i] based on digital (25, 35)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 25 and N(F) ≥ 36, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(72−47, 72, 85)-Net over F₃ — Upper bound on s (digital)

There is no digital (25, 72, 86)-net over F₃, because

1 times m-reduction [i] would yield digital (25, 71, 86)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁷¹, 86, F₃, 46) (dual of [86, 15, 47]-code), but
  - â€œDaâ€ bound on codes from Brouwerâ€™s database [i]

(72−47, 72, 89)-Net in Base 3 — Upper bound on s

There is no (25, 72, 90)-net in base 3, because

1 times m-reduction [i] would yield (25, 71, 90)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁷¹, 90, S₃, 46), but
  - the linear programming bound shows that MÂ â‰¥ 1 204855 461369 406663 581979 172263 433095 766907 / 144 201499 > 3⁷¹ [i]