MinT - Best Known (218−115, 218, s)-Nets in Base 4

Best Known (218−115, 218, s)-Nets in Base 4

(218−115, 218, 104)-Net over F₄ — Constructive and digital

Digital (103, 218, 104)-net over F₄, using

t-expansion [i] based on digital (73, 218, 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]

(218−115, 218, 144)-Net over F₄ — Digital

Digital (103, 218, 144)-net over F₄, using

t-expansion [i] based on digital (91, 218, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(218−115, 218, 1395)-Net in Base 4 — Upper bound on s

There is no (103, 218, 1396)-net in base 4, because

1 times m-reduction [i] would yield (103, 217, 1396)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45339 247078 622662 598315 389108 159135 092089 845787 393148 278820 766215 343155 290779 494463 588872 676084 813851 501557 061797 673092 557955 774296 > 4²¹⁷ [i]