MinT - Best Known (179−111, 179, s)-Nets in Base 4

Best Known (179−111, 179, s)-Nets in Base 4

(179−111, 179, 66)-Net over F₄ — Constructive and digital

Digital (68, 179, 66)-net over F₄, using

t-expansion [i] based on digital (49, 179, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(179−111, 179, 99)-Net over F₄ — Digital

Digital (68, 179, 99)-net over F₄, using

t-expansion [i] based on digital (61, 179, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(179−111, 179, 587)-Net in Base 4 — Upper bound on s

There is no (68, 179, 588)-net in base 4, because

1 times m-reduction [i] would yield (68, 178, 588)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 152246 198385 060879 120999 593816 716410 521224 342849 285647 836410 413392 992552 031222 936478 126758 890058 962549 011880 > 4¹⁷⁸ [i]