MinT - Best Known (211−129, 211, s)-Nets in Base 4

Best Known (211−129, 211, s)-Nets in Base 4

(211−129, 211, 104)-Net over F₄ — Constructive and digital

Digital (82, 211, 104)-net over F₄, using

t-expansion [i] based on digital (73, 211, 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]

(211−129, 211, 129)-Net over F₄ — Digital

Digital (82, 211, 129)-net over F₄, using

t-expansion [i] based on digital (81, 211, 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]

(211−129, 211, 725)-Net in Base 4 — Upper bound on s

There is no (82, 211, 726)-net in base 4, because

1 times m-reduction [i] would yield (82, 210, 726)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 738455 218887 911122 260056 157660 785240 036155 219442 355628 366163 761620 385100 430378 195718 499504 231882 637763 101416 310248 598272 152811 > 4²¹⁰ [i]