MinT - Best Known (66, 66+123, s)-Nets in Base 4

Best Known (66, 66+123, s)-Nets in Base 4

(66, 66+123, 66)-Net over F₄ — Constructive and digital

Digital (66, 189, 66)-net over F₄, using

t-expansion [i] based on digital (49, 189, 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]

(66, 66+123, 99)-Net over F₄ — Digital

Digital (66, 189, 99)-net over F₄, using

t-expansion [i] based on digital (61, 189, 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]

(66, 66+123, 514)-Net in Base 4 — Upper bound on s

There is no (66, 189, 515)-net in base 4, because

1 times m-reduction [i] would yield (66, 188, 515)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 163111 519407 546852 566741 449874 409624 246570 677272 871024 293592 163216 096681 550251 706323 481055 583071 215307 520063 056128 > 4¹⁸⁸ [i]