MinT - Best Known (210−173, 210, s)-Nets in Base 4

Best Known (210−173, 210, s)-Nets in Base 4

(210−173, 210, 56)-Net over F₄ — Constructive and digital

Digital (37, 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]

(210−173, 210, 66)-Net over F₄ — Digital

Digital (37, 210, 66)-net over F₄, using

net from sequence [i] based on digital (37, 65)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
  - Drinfeld modules of rank 1 [i]

(210−173, 210, 156)-Net over F₄ — Upper bound on s (digital)

There is no digital (37, 210, 157)-net over F₄, because

61 times m-reduction [i] would yield digital (37, 149, 157)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁹, 157, F₄, 112) (dual of [157, 8, 113]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁴⁸, 153, F₄, 112) (dual of [153, 5, 113]-code), but
      - residual code [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]
    2. OA(4⁸, 157, S₄, 4), but
      - discarding factors would yield OA(4⁸, 121, S₄, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 65704 > 4⁸ [i]

(210−173, 210, 161)-Net in Base 4 — Upper bound on s

There is no (37, 210, 162)-net in base 4, because

52 times m-reduction [i] would yield (37, 158, 162)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁵⁸, 162, S₄, 121), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 8 543948 143683 640329 580086 824678 208458 410818 089426 611079 788166 431288 878903 122562 200091 848347 746304 / 61 > 4¹⁵⁸ [i]