MinT - Best Known (39, 39+118, s)-Nets in Base 4

Best Known (39, 39+118, s)-Nets in Base 4

(39, 39+118, 56)-Net over F₄ — Constructive and digital

Digital (39, 157, 56)-net over F₄, using

t-expansion [i] based on digital (33, 157, 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+118, 66)-Net over F₄ — Digital

Digital (39, 157, 66)-net over F₄, using

t-expansion [i] based on digital (37, 157, 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+118, 169)-Net over F₄ — Upper bound on s (digital)

There is no digital (39, 157, 170)-net over F₄, because

2 times m-reduction [i] would yield digital (39, 155, 170)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁵⁵, 170, F₄, 116) (dual of [170, 15, 117]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4¹⁵⁴, 162, F₄, 116) (dual of [162, 8, 117]-code), but
      - constructionÂ Y1 [i] would yield
        linear OA(4¹⁵³, 158, F₄, 116) (dual of [158, 5, 117]-code), but
        residual code [i] would yield linear OA(4³⁷, 41, F₄, 29) (dual of [41, 4, 30]-code), but
        1 times truncation [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, 162, 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]
    2. OA(4¹⁵, 170, 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]

(39, 39+118, 258)-Net in Base 4 — Upper bound on s

There is no (39, 157, 259)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 37610 057867 805820 288930 861655 450677 759999 094034 011494 195482 320140 091844 501634 784191 018395 960448 > 4¹⁵⁷ [i]