MinT - Best Known (240−155, 240, s)-Nets in Base 4

Best Known (240−155, 240, s)-Nets in Base 4

(240−155, 240, 104)-Net over F₄ — Constructive and digital

Digital (85, 240, 104)-net over F₄, using

t-expansion [i] based on digital (73, 240, 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]

(240−155, 240, 129)-Net over F₄ — Digital

Digital (85, 240, 129)-net over F₄, using

t-expansion [i] based on digital (81, 240, 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]

(240−155, 240, 664)-Net in Base 4 — Upper bound on s

There is no (85, 240, 665)-net in base 4, because

1 times m-reduction [i] would yield (85, 239, 665)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 789998 740358 219990 176338 132275 033060 809376 061564 821120 825612 939275 257388 125301 752777 910493 000182 686767 908178 541801 534130 765243 993204 408780 955968 > 4²³⁹ [i]