MinT - Best Known (39, 39+113, s)-Nets in Base 4

Best Known (39, 39+113, s)-Nets in Base 4

(39, 39+113, 56)-Net over F₄ — Constructive and digital

Digital (39, 152, 56)-net over F₄, using

t-expansion [i] based on digital (33, 152, 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]

(39, 39+113, 66)-Net over F₄ — Digital

Digital (39, 152, 66)-net over F₄, using

t-expansion [i] based on digital (37, 152, 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]

(39, 39+113, 184)-Net over F₄ — Upper bound on s (digital)

There is no digital (39, 152, 185)-net over F₄, because

1 times m-reduction [i] would yield digital (39, 151, 185)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁵¹, 185, F₄, 112) (dual of [185, 34, 113]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁵⁰, 165, F₄, 112) (dual of [165, 15, 113]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(4¹⁴⁹, 157, F₄, 112) (dual of [157, 8, 113]-code), but
        constructionÂ Y1 [i] would yield
        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]
        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]
        OA(4¹⁵, 165, S₄, 8), but
        discarding factors would yield OA(4¹⁵, 135, S₄, 8), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1082 768311 > 4¹⁵ [i]
    2. OA(4³⁴, 185, S₄, 20), but
      - discarding factors would yield OA(4³⁴, 173, S₄, 20), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 305 996090 287239 486286 > 4³⁴ [i]

(39, 39+113, 260)-Net in Base 4 — Upper bound on s

There is no (39, 152, 261)-net in base 4, because

1 times m-reduction [i] would yield (39, 151, 261)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 174740 279627 657992 909585 340546 074555 658524 412095 078813 567698 619201 938737 344811 195323 686880 > 4¹⁵¹ [i]