MinT - Best Known (11, 45, s)-Nets in Base 4

Best Known (11, 45, s)-Nets in Base 4

(11, 45, 27)-Net over F₄ — Constructive and digital

Digital (11, 45, 27)-net over F₄, using

t-expansion [i] based on digital (10, 45, 27)-net over F₄, using
- net from sequence [i] based on digital (10, 26)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
    - an explicitly constructive algebraic function field [i]

(11, 45, 53)-Net over F₄ — Upper bound on s (digital)

There is no digital (11, 45, 54)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4⁴⁵, 54, F₄, 34) (dual of [54, 9, 35]-code), but
- â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(11, 45, 56)-Net in Base 4 — Upper bound on s

There is no (11, 45, 57)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁴⁵, 57, S₄, 34), but
- the linear programming bound shows that MÂ â‰¥ 91952 839239 355304 325767 857111 564288 / 50 096875 > 4⁴⁵ [i]