MinT - Best Known (210−163, 210, s)-Nets in Base 4

Best Known (210−163, 210, s)-Nets in Base 4

(210−163, 210, 56)-Net over F₄ — Constructive and digital

Digital (47, 210, 56)-net over F₄, using

t-expansion [i] based on digital (33, 210, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(210−163, 210, 81)-Net over F₄ — Digital

Digital (47, 210, 81)-net over F₄, using

t-expansion [i] based on digital (46, 210, 81)-net over F₄, using
- net from sequence [i] based on digital (46, 80)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 46 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(210−163, 210, 195)-Net over F₄ — Upper bound on s (digital)

There is no digital (47, 210, 196)-net over F₄, because

19 times m-reduction [i] would yield digital (47, 191, 196)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁹¹, 196, F₄, 144) (dual of [196, 5, 145]-code), but
  - residual code [i] would yield linear OA(4⁴⁷, 51, F₄, 36) (dual of [51, 4, 37]-code), but
    - a result by Landjev, Maruta, and Hill [i]

(210−163, 210, 202)-Net in Base 4 — Upper bound on s

There is no (47, 210, 203)-net in base 4, because

11 times m-reduction [i] would yield (47, 199, 203)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁹⁹, 203, S₄, 152), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 41 315998 049390 537434 494706 752048 189989 275292 685267 576205 290549 704650 361952 269459 114074 325652 482205 302974 450751 563959 894016 / 51 > 4¹⁹⁹ [i]