MinT - Best Known (247−163, 247, s)-Nets in Base 4

Best Known (247−163, 247, s)-Nets in Base 4

(247−163, 247, 104)-Net over F₄ — Constructive and digital

Digital (84, 247, 104)-net over F₄, using

t-expansion [i] based on digital (73, 247, 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]

(247−163, 247, 129)-Net over F₄ — Digital

Digital (84, 247, 129)-net over F₄, using

t-expansion [i] based on digital (81, 247, 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]

(247−163, 247, 630)-Net in Base 4 — Upper bound on s

There is no (84, 247, 631)-net in base 4, because

1 times m-reduction [i] would yield (84, 246, 631)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13109 927246 748492 847432 833529 464219 509427 486861 682090 422251 816363 662771 270964 810569 714124 192182 298669 101216 925949 860885 708576 618056 114518 279676 489582 > 4²⁴⁶ [i]