MinT - Best Known (194−129, 194, s)-Nets in Base 4

Best Known (194−129, 194, s)-Nets in Base 4

(194−129, 194, 66)-Net over F₄ — Constructive and digital

Digital (65, 194, 66)-net over F₄, using

t-expansion [i] based on digital (49, 194, 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]

(194−129, 194, 99)-Net over F₄ — Digital

Digital (65, 194, 99)-net over F₄, using

t-expansion [i] based on digital (61, 194, 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]

(194−129, 194, 486)-Net in Base 4 — Upper bound on s

There is no (65, 194, 487)-net in base 4, because

1 times m-reduction [i] would yield (65, 193, 487)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 158 389908 730260 678135 121901 397454 790714 279258 197997 071461 989075 031756 308044 306767 459550 395334 624128 962216 581485 300157 > 4¹⁹³ [i]