MinT - Best Known (217−137, 217, s)-Nets in Base 4

Best Known (217−137, 217, s)-Nets in Base 4

(217−137, 217, 104)-Net over F₄ — Constructive and digital

Digital (80, 217, 104)-net over F₄, using

t-expansion [i] based on digital (73, 217, 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]

(217−137, 217, 112)-Net over F₄ — Digital

Digital (80, 217, 112)-net over F₄, using

t-expansion [i] based on digital (73, 217, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(217−137, 217, 658)-Net in Base 4 — Upper bound on s

There is no (80, 217, 659)-net in base 4, because

1 times m-reduction [i] would yield (80, 216, 659)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12091 114897 304370 176682 956970 925566 345493 723378 443992 321397 584019 479279 358976 421571 753672 149270 292153 927942 010317 733574 815201 495000 > 4²¹⁶ [i]