MinT - Best Known (48, 48+209, s)-Nets in Base 4

Best Known (48, 48+209, s)-Nets in Base 4

(48, 48+209, 56)-Net over F₄ — Constructive and digital

Digital (48, 257, 56)-net over F₄, using

t-expansion [i] based on digital (33, 257, 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]

(48, 48+209, 81)-Net over F₄ — Digital

Digital (48, 257, 81)-net over F₄, using

t-expansion [i] based on digital (46, 257, 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]

(48, 48+209, 199)-Net over F₄ — Upper bound on s (digital)

There is no digital (48, 257, 200)-net over F₄, because

65 times m-reduction [i] would yield digital (48, 192, 200)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁹², 200, F₄, 144) (dual of [200, 8, 145]-code), but
  - constructionÂ Y1 [i] would yield
    1. 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]
    2. OA(4⁸, 200, 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]

(48, 48+209, 206)-Net in Base 4 — Upper bound on s

There is no (48, 257, 207)-net in base 4, because

54 times m-reduction [i] would yield (48, 203, 207)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4²⁰³, 207, S₄, 155), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 2644 223875 160994 395807 661232 131084 159313 618731 857124 877138 595181 097623 164945 245383 300756 841758 861139 390364 848100 093433 217024 / 13 > 4²⁰³ [i]