MinT - Best Known (121−77, 121, s)-Nets in Base 4

Best Known (121−77, 121, s)-Nets in Base 4

(121−77, 121, 56)-Net over F₄ — Constructive and digital

Digital (44, 121, 56)-net over F₄, using

t-expansion [i] based on digital (33, 121, 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]

(121−77, 121, 75)-Net over F₄ — Digital

Digital (44, 121, 75)-net over F₄, using

t-expansion [i] based on digital (40, 121, 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]

(121−77, 121, 368)-Net in Base 4 — Upper bound on s

There is no (44, 121, 369)-net in base 4, because

1 times m-reduction [i] would yield (44, 120, 369)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 820638 386243 423675 995082 742890 257779 154457 268749 285730 233460 540995 137320 > 4¹²⁰ [i]