MinT - Best Known (42, 165, s)-Nets in Base 4

Best Known (42, 165, s)-Nets in Base 4

Digital (42, 165, 56)-net over F₄, using

t-expansion [i] based on digital (33, 165, 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 (42, 165, 75)-net over F₄, using

t-expansion [i] based on digital (40, 165, 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 (42, 165, 209)-net over F₄, because

3 times m-reduction [i] would yield digital (42, 162, 209)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁶², 209, F₄, 120) (dual of [209, 47, 121]-code), but
  - residual code [i] would yield OA(4⁴², 88, S₄, 30), but
    - the linear programming bound shows that MÂ â‰¥ 89 605050 492137 134852 017163 586838 439030 395196 856884 186345 777644 655144 707653 985466 546419 715121 697993 523200 / 4 627392 744873 205288 863270 778823 516250 097275 954094 267550 824480 805099 989747 440773 > 4⁴² [i]

There is no (42, 165, 279)-net in base 4, because

1 times m-reduction [i] would yield (42, 164, 279)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 565 966260 284655 585061 969286 398178 288280 961870 703705 135911 058800 420827 149997 661122 819652 703823 766720 > 4¹⁶⁴ [i]