MinT - Best Known (123−63, 123, s)-Nets in Base 4

Best Known (123−63, 123, s)-Nets in Base 4

(123−63, 123, 66)-Net over F₄ — Constructive and digital

Digital (60, 123, 66)-net over F₄, using

t-expansion [i] based on digital (49, 123, 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]

(123−63, 123, 91)-Net over F₄ — Digital

Digital (60, 123, 91)-net over F₄, using

t-expansion [i] based on digital (50, 123, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(123−63, 123, 943)-Net in Base 4 — Upper bound on s

There is no (60, 123, 944)-net in base 4, because

1 times m-reduction [i] would yield (60, 122, 944)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 28 373769 925420 762807 487876 479854 849446 446733 080306 828209 502648 769097 591070 > 4¹²² [i]