MinT - Best Known (5, 20, s)-Nets in Base 4

Best Known (5, 20, s)-Nets in Base 4

(5, 20, 17)-Net over F₄ — Constructive and digital

Digital (5, 20, 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, 20, 39)-Net over F₄ — Upper bound on s (digital)

There is no digital (5, 20, 40)-net over F₄, because

1 times m-reduction [i] would yield digital (5, 19, 40)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁹, 40, F₄, 14) (dual of [40, 21, 15]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁸, 24, F₄, 14) (dual of [24, 6, 15]-code), but
      - â€œBouâ€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(4²¹, 40, F₄, 16) (dual of [40, 19, 17]-code), but
      - discarding factors / shortening the dual code would yield linear OA(4²¹, 32, F₄, 16) (dual of [32, 11, 17]-code), but
        residual code [i] would yield OA(4⁵, 15, S₄, 4), but
        the linear programming bound shows that MÂ â‰¥ 3328 / 3 > 4⁵ [i]

(5, 20, 43)-Net in Base 4 — Upper bound on s

There is no (5, 20, 44)-net in base 4, because

1 times m-reduction [i] would yield (5, 19, 44)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 299064 681070 > 4¹⁹ [i]