MinT - Best Known (126−78, 126, s)-Nets in Base 4

Best Known (126−78, 126, s)-Nets in Base 4

(126−78, 126, 56)-Net over F₄ — Constructive and digital

Digital (48, 126, 56)-net over F₄, using

t-expansion [i] based on digital (33, 126, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(126−78, 126, 81)-Net over F₄ — Digital

Digital (48, 126, 81)-net over F₄, using

t-expansion [i] based on digital (46, 126, 81)-net over F₄, using
- net from sequence [i] based on digital (46, 80)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 46 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(126−78, 126, 421)-Net in Base 4 — Upper bound on s

There is no (48, 126, 422)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7800 455347 599757 271784 372438 979200 118923 000457 126200 283945 139071 553453 474608 > 4¹²⁶ [i]