MinT - Best Known (164−101, 164, s)-Nets in Base 4

Best Known (164−101, 164, s)-Nets in Base 4

(164−101, 164, 66)-Net over F₄ — Constructive and digital

Digital (63, 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]

(164−101, 164, 99)-Net over F₄ — Digital

Digital (63, 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]

(164−101, 164, 555)-Net in Base 4 — Upper bound on s

There is no (63, 164, 556)-net in base 4, because

1 times m-reduction [i] would yield (63, 163, 556)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 137 899880 916262 533254 322651 075497 923441 347390 740706 265054 966810 882302 197974 163692 176259 987459 045408 > 4¹⁶³ [i]