MinT - Best Known (185−117, 185, s)-Nets in Base 4

Best Known (185−117, 185, s)-Nets in Base 4

(185−117, 185, 66)-Net over F₄ — Constructive and digital

Digital (68, 185, 66)-net over F₄, using

t-expansion [i] based on digital (49, 185, 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]

(185−117, 185, 99)-Net over F₄ — Digital

Digital (68, 185, 99)-net over F₄, using

t-expansion [i] based on digital (61, 185, 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]

(185−117, 185, 561)-Net in Base 4 — Upper bound on s

There is no (68, 185, 562)-net in base 4, because

1 times m-reduction [i] would yield (68, 184, 562)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 612 394064 830502 170692 327497 375787 341633 589991 943344 307584 681250 677937 285053 543384 282150 981242 035249 973717 387520 > 4¹⁸⁴ [i]