MinT - Best Known (57−50, 57, s)-Nets in Base 4

Best Known (57−50, 57, s)-Nets in Base 4

(57−50, 57, 21)-Net over F₄ — Constructive and digital

Digital (7, 57, 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]

(57−50, 57, 34)-Net over F₄ — Upper bound on s (digital)

There is no digital (7, 57, 35)-net over F₄, because

26 times m-reduction [i] would yield digital (7, 31, 35)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4³¹, 35, F₄, 24) (dual of [35, 4, 25]-code), but
  - residual code [i] would yield linear OA(4⁷, 10, F₄, 6) (dual of [10, 3, 7]-code), but
    - bound for almost-MDS-codes with dimension nÂ =Â 3 [i]

(57−50, 57, 37)-Net in Base 4 — Upper bound on s

There is no (7, 57, 38)-net in base 4, because

25 times m-reduction [i] would yield (7, 32, 38)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4³², 38, S₄, 25), but
  - the linear programming bound shows that MÂ â‰¥ 7821 419487 252849 885184 / 403 > 4³² [i]