MinT - Best Known (171−105, 171, s)-Nets in Base 4

Best Known (171−105, 171, s)-Nets in Base 4

(171−105, 171, 66)-Net over F₄ — Constructive and digital

Digital (66, 171, 66)-net over F₄, using

t-expansion [i] based on digital (49, 171, 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]

(171−105, 171, 99)-Net over F₄ — Digital

Digital (66, 171, 99)-net over F₄, using

t-expansion [i] based on digital (61, 171, 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]

(171−105, 171, 584)-Net in Base 4 — Upper bound on s

There is no (66, 171, 585)-net in base 4, because

1 times m-reduction [i] would yield (66, 170, 585)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 252432 152384 834766 055186 215737 389933 112488 554923 177371 163763 033842 174071 748202 706781 645048 570740 681640 > 4¹⁷⁰ [i]