MinT - Best Known (3, 3+13, s)-Nets in Base 4

Best Known (3, 3+13, s)-Nets in Base 4

(3, 3+13, 14)-Net over F₄ — Constructive and digital

Digital (3, 16, 14)-net over F₄, using

net from sequence [i] based on digital (3, 13)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
  - an explicitly constructive algebraic function field [i]

(3, 3+13, 17)-Net over F₄ — Upper bound on s (digital)

There is no digital (3, 16, 18)-net over F₄, because

3 times m-reduction [i] would yield digital (3, 13, 18)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹³, 18, F₄, 10) (dual of [18, 5, 11]-code), but
  - â€œLizâ€ bound on codes from Brouwerâ€™s database [i]

(3, 3+13, 20)-Net in Base 4 — Upper bound on s

There is no (3, 16, 21)-net in base 4, because

extracting embedded OOA [i] would yield OOA(4¹⁶, 21, S₄, 2, 13), but
- the linear programming bound for OOAs shows that MÂ â‰¥ 202192 225478 639616 / 44 300929 > 4¹⁶ [i]