MinT - Best Known (160−119, 160, s)-Nets in Base 4

Best Known (160−119, 160, s)-Nets in Base 4

Digital (41, 160, 56)-net over F₄, using

t-expansion [i] based on digital (33, 160, 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]

Digital (41, 160, 75)-net over F₄, using

t-expansion [i] based on digital (40, 160, 75)-net over F₄, using
- net from sequence [i] based on digital (40, 74)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
    - Drinfeld modules of rank 1 [i]

There is no digital (41, 160, 209)-net over F₄, because

3 times m-reduction [i] would yield digital (41, 157, 209)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁵⁷, 209, F₄, 116) (dual of [209, 52, 117]-code), but
  - residual code [i] would yield OA(4⁴¹, 92, S₄, 29), but
    - the linear programming bound shows that MÂ â‰¥ 8291 138749 676505 103841 959950 541753 992228 411787 128227 036594 724129 251790 946304 / 1649 446049 086149 969896 114658 918272 865279 146667 501047 > 4⁴¹ [i]

There is no (41, 160, 273)-net in base 4, because

1 times m-reduction [i] would yield (41, 159, 273)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 542788 104776 016167 487161 968872 148714 513851 125132 060833 580944 245976 991177 893347 486816 309706 007296 > 4¹⁵⁹ [i]