MinT - Best Known (239−190, 239, s)-Nets in Base 4

Best Known (239−190, 239, s)-Nets in Base 4

(239−190, 239, 66)-Net over F₄ — Constructive and digital

Digital (49, 239, 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]

(239−190, 239, 81)-Net over F₄ — Digital

Digital (49, 239, 81)-net over F₄, using

t-expansion [i] based on digital (46, 239, 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]

(239−190, 239, 204)-Net over F₄ — Upper bound on s (digital)

There is no digital (49, 239, 205)-net over F₄, because

42 times m-reduction [i] would yield digital (49, 197, 205)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁹⁷, 205, F₄, 148) (dual of [205, 8, 149]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁹⁶, 201, F₄, 148) (dual of [201, 5, 149]-code), but
      - residual code [i] would yield linear OA(4⁴⁸, 52, F₄, 37) (dual of [52, 4, 38]-code), but
        1 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, 205, 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]

(239−190, 239, 210)-Net in Base 4 — Upper bound on s

There is no (49, 239, 211)-net in base 4, because

32 times m-reduction [i] would yield (49, 207, 211)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4²⁰⁷, 211, S₄, 158), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 2 707685 248164 858261 307045 101702 230179 137145 581421 695874 189921 465443 966120 903931 272499 975005 961073 806735 733604 454495 675614 232576 / 53 > 4²⁰⁷ [i]