MinT - Best Known (164−97, 164, s)-Nets in Base 4

Best Known (164−97, 164, s)-Nets in Base 4

(164−97, 164, 66)-Net over F₄ — Constructive and digital

Digital (67, 164, 66)-net over F₄, using

t-expansion [i] based on digital (49, 164, 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]

(164−97, 164, 99)-Net over F₄ — Digital

Digital (67, 164, 99)-net over F₄, using

t-expansion [i] based on digital (61, 164, 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]

(164−97, 164, 653)-Net in Base 4 — Upper bound on s

There is no (67, 164, 654)-net in base 4, because

1 times m-reduction [i] would yield (67, 163, 654)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 142 222234 304524 830029 772364 238872 542651 141295 721167 304000 191861 744806 496220 549451 698536 687889 574904 > 4¹⁶³ [i]