MinT - Best Known (232−145, 232, s)-Nets in Base 4

Best Known (232−145, 232, s)-Nets in Base 4

(232−145, 232, 104)-Net over F₄ — Constructive and digital

Digital (87, 232, 104)-net over F₄, using

t-expansion [i] based on digital (73, 232, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(232−145, 232, 129)-Net over F₄ — Digital

Digital (87, 232, 129)-net over F₄, using

t-expansion [i] based on digital (81, 232, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(232−145, 232, 729)-Net in Base 4 — Upper bound on s

There is no (87, 232, 730)-net in base 4, because

1 times m-reduction [i] would yield (87, 231, 730)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 873114 665614 599242 355351 506159 679312 380743 901661 438566 668102 075058 117264 669807 068929 669272 257310 481718 012930 226113 995210 087616 159514 851741 > 4²³¹ [i]