MinT - Best Known (51, 51+194, s)-Nets in Base 4

Best Known (51, 51+194, s)-Nets in Base 4

(51, 51+194, 66)-Net over F₄ — Constructive and digital

Digital (51, 245, 66)-net over F₄, using

t-expansion [i] based on digital (49, 245, 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]

(51, 51+194, 91)-Net over F₄ — Digital

Digital (51, 245, 91)-net over F₄, using

t-expansion [i] based on digital (50, 245, 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]

(51, 51+194, 214)-Net over F₄ — Upper bound on s (digital)

There is no digital (51, 245, 215)-net over F₄, because

38 times m-reduction [i] would yield digital (51, 207, 215)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁷, 215, F₄, 156) (dual of [215, 8, 157]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4²⁰⁶, 211, F₄, 156) (dual of [211, 5, 157]-code), but
      - residual code [i] would yield linear OA(4⁵⁰, 54, F₄, 39) (dual of [54, 4, 40]-code), but
        3 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, 215, 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]

(51, 51+194, 218)-Net in Base 4 — Upper bound on s

There is no (51, 245, 219)-net in base 4, because

30 times m-reduction [i] would yield (51, 215, 219)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4²¹⁵, 219, S₄, 164), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 177450 860423 732151 013018 507785 157357 019931 972824 052260 810910 693159 335763 699560 039874 558361 990664 932998 233037 501529 828597 054346 100736 / 55 > 4²¹⁵ [i]