MinT - Best Known (180−123, 180, s)-Nets in Base 4

Best Known (180−123, 180, s)-Nets in Base 4

(180−123, 180, 66)-Net over F₄ — Constructive and digital

Digital (57, 180, 66)-net over F₄, using

t-expansion [i] based on digital (49, 180, 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]

(180−123, 180, 91)-Net over F₄ — Digital

Digital (57, 180, 91)-net over F₄, using

t-expansion [i] based on digital (50, 180, 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]

(180−123, 180, 410)-Net in Base 4 — Upper bound on s

There is no (57, 180, 411)-net in base 4, because

1 times m-reduction [i] would yield (57, 179, 411)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 615267 693166 477955 191855 968056 716082 673392 205859 920502 412503 302205 669788 053944 639816 657999 764999 525740 703040 > 4¹⁷⁹ [i]