MinT - Best Known (15, 60, s)-Nets in Base 4

Best Known (15, 60, s)-Nets in Base 4

(15, 60, 33)-Net over F₄ — Constructive and digital

Digital (15, 60, 33)-net over F₄, using

net from sequence [i] based on digital (15, 32)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
  - an explicitly constructive algebraic function field [i]

(15, 60, 35)-Net over F₄ — Digital

Digital (15, 60, 35)-net over F₄, using

net from sequence [i] based on digital (15, 34)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 35, using
  - a curve from the manYPoints database [i]

(15, 60, 70)-Net over F₄ — Upper bound on s (digital)

There is no digital (15, 60, 71)-net over F₄, because

1 times m-reduction [i] would yield digital (15, 59, 71)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁵⁹, 71, F₄, 44) (dual of [71, 12, 45]-code), but
  - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(15, 60, 71)-Net in Base 4 — Upper bound on s

There is no (15, 60, 72)-net in base 4, because

1 times m-reduction [i] would yield (15, 59, 72)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁵⁹, 72, S₄, 44), but
  - the linear programming bound shows that MÂ â‰¥ 3 514436 285559 452450 649732 945555 302087 917568 / 7 940309 > 4⁵⁹ [i]