MinT - Best Known (56, 56+82, s)-Nets in Base 4

Best Known (56, 56+82, s)-Nets in Base 4

(56, 56+82, 66)-Net over F₄ — Constructive and digital

Digital (56, 138, 66)-net over F₄, using

t-expansion [i] based on digital (49, 138, 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]

(56, 56+82, 91)-Net over F₄ — Digital

Digital (56, 138, 91)-net over F₄, using

t-expansion [i] based on digital (50, 138, 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]

(56, 56+82, 538)-Net in Base 4 — Upper bound on s

There is no (56, 138, 539)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 122630 398726 382181 067549 012816 413598 911022 802989 310372 236550 661033 724639 607480 992480 > 4¹³⁸ [i]