MinT - Best Known (68, 68+89, s)-Nets in Base 4

Best Known (68, 68+89, s)-Nets in Base 4

(68, 68+89, 66)-Net over F₄ — Constructive and digital

Digital (68, 157, 66)-net over F₄, using

t-expansion [i] based on digital (49, 157, 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]

(68, 68+89, 99)-Net over F₄ — Digital

Digital (68, 157, 99)-net over F₄, using

t-expansion [i] based on digital (61, 157, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(68, 68+89, 748)-Net in Base 4 — Upper bound on s

There is no (68, 157, 749)-net in base 4, because

1 times m-reduction [i] would yield (68, 156, 749)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8548 290754 249365 808232 310829 442159 541564 073990 323672 458938 923609 023481 323274 279879 539227 307779 > 4¹⁵⁶ [i]