MinT - Best Known (50, 50+165, s)-Nets in Base 4

Best Known (50, 50+165, s)-Nets in Base 4

(50, 50+165, 66)-Net over F₄ — Constructive and digital

Digital (50, 215, 66)-net over F₄, using

t-expansion [i] based on digital (49, 215, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(50, 50+165, 91)-Net over F₄ — Digital

Digital (50, 215, 91)-net over F₄, using

net from sequence [i] based on digital (50, 90)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
  - a curve from the manYPoints database [i]

(50, 50+165, 209)-Net over F₄ — Upper bound on s (digital)

There is no digital (50, 215, 210)-net over F₄, because

13 times m-reduction [i] would yield digital (50, 202, 210)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰², 210, F₄, 152) (dual of [210, 8, 153]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4²⁰¹, 206, F₄, 152) (dual of [206, 5, 153]-code), but
      - residual code [i] would yield linear OA(4⁴⁹, 53, F₄, 38) (dual of [53, 4, 39]-code), but
        2 times truncation [i] would yield linear OA(4⁴⁷, 51, F₄, 36) (dual of [51, 4, 37]-code), but
        a result by Landjev, Maruta, and Hill [i]
    2. OA(4⁸, 210, 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]

(50, 50+165, 214)-Net in Base 4 — Upper bound on s

There is no (50, 215, 215)-net in base 4, because

4 times m-reduction [i] would yield (50, 211, 215)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4²¹¹, 215, S₄, 161), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 346 583711 765101 857447 301773 017885 462929 554634 421977 071896 309947 576827 663475 703202 879996 800763 017447 262173 901370 175446 478621 769728 / 27 > 4²¹¹ [i]