MinT - Best Known (50−42, 50, s)-Nets in Base 4

Best Known (50−42, 50, s)-Nets in Base 4

(50−42, 50, 21)-Net over F₄ — Constructive and digital

Digital (8, 50, 21)-net over F₄, using

t-expansion [i] based on digital (7, 50, 21)-net over F₄, using
- net from sequence [i] based on digital (7, 20)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(50−42, 50, 39)-Net over F₄ — Upper bound on s (digital)

There is no digital (8, 50, 40)-net over F₄, because

14 times m-reduction [i] would yield digital (8, 36, 40)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4³⁶, 40, F₄, 28) (dual of [40, 4, 29]-code), but
  - residual code [i] would yield linear OA(4⁸, 11, F₄, 7) (dual of [11, 3, 8]-code), but
    - improvement by Maruta on the Griesmer bound [i]

(50−42, 50, 41)-Net in Base 4 — Upper bound on s

There is no (8, 50, 42)-net in base 4, because

14 times m-reduction [i] would yield (8, 36, 42)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4³⁶, 42, S₄, 28), but
  - the linear programming bound shows that MÂ â‰¥ 13 298184 015760 920921 767936 / 2465 > 4³⁶ [i]