MinT - Best Known (42, 42+119, s)-Nets in Base 4

Best Known (42, 42+119, s)-Nets in Base 4

(42, 42+119, 56)-Net over F₄ — Constructive and digital

Digital (42, 161, 56)-net over F₄, using

t-expansion [i] based on digital (33, 161, 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]

(42, 42+119, 75)-Net over F₄ — Digital

Digital (42, 161, 75)-net over F₄, using

t-expansion [i] based on digital (40, 161, 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]

(42, 42+119, 222)-Net over F₄ — Upper bound on s (digital)

There is no digital (42, 161, 223)-net over F₄, because

3 times m-reduction [i] would yield digital (42, 158, 223)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁵⁸, 223, F₄, 116) (dual of [223, 65, 117]-code), but
  - residual code [i] would yield OA(4⁴², 106, S₄, 29), but
    - the linear programming bound shows that MÂ â‰¥ 11 515430 185021 615009 307015 123093 343331 130678 608727 482845 042118 230016 / 573326 282922 594402 886717 422034 435890 047071 > 4⁴² [i]

(42, 42+119, 280)-Net in Base 4 — Upper bound on s

There is no (42, 161, 281)-net in base 4, because

1 times m-reduction [i] would yield (42, 160, 281)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 365727 016581 348266 231073 212536 314075 007315 277637 982302 991325 864699 097104 277630 290241 007959 607040 > 4¹⁶⁰ [i]