MinT - Best Known (5, 25, s)-Nets in Base 4

Best Known (5, 25, s)-Nets in Base 4

(5, 25, 17)-Net over F₄ — Constructive and digital

Digital (5, 25, 17)-net over F₄, using

net from sequence [i] based on digital (5, 16)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
  - an explicitly constructive algebraic function field [i]

(5, 25, 26)-Net over F₄ — Upper bound on s (digital)

There is no digital (5, 25, 27)-net over F₄, because

3 times m-reduction [i] would yield digital (5, 22, 27)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²², 27, F₄, 17) (dual of [27, 5, 18]-code), but
  - â€œBouâ€ bound on codes from Brouwerâ€™s database [i]

(5, 25, 29)-Net in Base 4 — Upper bound on s

There is no (5, 25, 30)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4²⁵, 30, S₄, 20), but
- the linear programming bound shows that MÂ â‰¥ 576460 752303 423488 / 455 > 4²⁵ [i]