MinT - Best Known (75−61, 75, s)-Nets in Base 3

Best Known (75−61, 75, s)-Nets in Base 3

(75−61, 75, 24)-Net over F₃ — Constructive and digital

Digital (14, 75, 24)-net over F₃, using

t-expansion [i] based on digital (13, 75, 24)-net over F₃, using
- net from sequence [i] based on digital (13, 23)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
    - a function field by SÃ©mirat [i]

(75−61, 75, 49)-Net over F₃ — Upper bound on s (digital)

There is no digital (14, 75, 50)-net over F₃, because

31 times m-reduction [i] would yield digital (14, 44, 50)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁴⁴, 50, F₃, 30) (dual of [50, 6, 31]-code), but
  - â€œHJLâ€ bound on codes from Brouwerâ€™s database [i]

(75−61, 75, 51)-Net in Base 3 — Upper bound on s

There is no (14, 75, 52)-net in base 3, because

30 times m-reduction [i] would yield (14, 45, 52)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁴⁵, 52, S₃, 31), but
  - the linear programming bound shows that MÂ â‰¥ 239299 329230 617529 590083 / 76 > 3⁴⁵ [i]