MinT - Best Known (228−135, 228, s)-Nets in Base 4

Best Known (228−135, 228, s)-Nets in Base 4

(228−135, 228, 104)-Net over F₄ — Constructive and digital

Digital (93, 228, 104)-net over F₄, using

t-expansion [i] based on digital (73, 228, 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]

(228−135, 228, 144)-Net over F₄ — Digital

Digital (93, 228, 144)-net over F₄, using

t-expansion [i] based on digital (91, 228, 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]

(228−135, 228, 887)-Net in Base 4 — Upper bound on s

There is no (93, 228, 888)-net in base 4, because

1 times m-reduction [i] would yield (93, 227, 888)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46837 161425 756430 436696 727049 945547 361303 419375 153405 370963 950695 919241 690551 184291 623677 025742 152302 091392 694771 817132 793775 082392 111555 > 4²²⁷ [i]