MinT - Best Known (248−165, 248, s)-Nets in Base 4

Best Known (248−165, 248, s)-Nets in Base 4

(248−165, 248, 104)-Net over F₄ — Constructive and digital

Digital (83, 248, 104)-net over F₄, using

t-expansion [i] based on digital (73, 248, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(248−165, 248, 129)-Net over F₄ — Digital

Digital (83, 248, 129)-net over F₄, using

t-expansion [i] based on digital (81, 248, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(248−165, 248, 614)-Net in Base 4 — Upper bound on s

There is no (83, 248, 615)-net in base 4, because

1 times m-reduction [i] would yield (83, 247, 615)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 54029 011158 091788 190491 676146 354302 028715 235010 030577 292996 794344 965512 856589 586735 894275 389761 323990 538818 442181 084716 496365 840870 834145 380659 390070 > 4²⁴⁷ [i]