MinT - Best Known (51−39, 51, s)-Nets in Base 4

Best Known (51−39, 51, s)-Nets in Base 4

(51−39, 51, 28)-Net over F₄ — Constructive and digital

Digital (12, 51, 28)-net over F₄, using

net from sequence [i] based on digital (12, 27)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
  - an explicitly constructive algebraic function field [i]

(51−39, 51, 29)-Net over F₄ — Digital

Digital (12, 51, 29)-net over F₄, using

net from sequence [i] based on digital (12, 28)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 29, using
  - a curve from the manYPoints database [i]

(51−39, 51, 56)-Net over F₄ — Upper bound on s (digital)

There is no digital (12, 51, 57)-net over F₄, because

3 times m-reduction [i] would yield digital (12, 48, 57)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4⁴⁸, 57, F₄, 36) (dual of [57, 9, 37]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4⁴⁷, 51, F₄, 36) (dual of [51, 4, 37]-code), but
      - a result by Landjev, Maruta, and Hill [i]
    2. OA(4⁹, 57, S₄, 6), but
      - discarding factors would yield OA(4⁹, 40, S₄, 6), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 273901 > 4⁹ [i]

(51−39, 51, 58)-Net in Base 4 — Upper bound on s

There is no (12, 51, 59)-net in base 4, because

1 times m-reduction [i] would yield (12, 50, 59)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁵⁰, 59, S₄, 38), but
  - the linear programming bound shows that MÂ â‰¥ 12 890525 988420 026441 280523 449330 040832 / 8 204625 > 4⁵⁰ [i]