MinT - Best Known (218−147, 218, s)-Nets in Base 4

Best Known (218−147, 218, s)-Nets in Base 4

(218−147, 218, 66)-Net over F₄ — Constructive and digital

Digital (71, 218, 66)-net over F₄, using

t-expansion [i] based on digital (49, 218, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(218−147, 218, 105)-Net over F₄ — Digital

Digital (71, 218, 105)-net over F₄, using

t-expansion [i] based on digital (70, 218, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(218−147, 218, 517)-Net in Base 4 — Upper bound on s

There is no (71, 218, 518)-net in base 4, because

1 times m-reduction [i] would yield (71, 217, 518)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 49310 236835 294339 347243 861131 125471 143814 148734 786722 092559 452671 926531 000401 901431 942869 765234 482189 773606 559568 963082 092979 966400 > 4²¹⁷ [i]