MinT - Best Known (170−107, 170, s)-Nets in Base 4

Best Known (170−107, 170, s)-Nets in Base 4

(170−107, 170, 66)-Net over F₄ — Constructive and digital

Digital (63, 170, 66)-net over F₄, using

t-expansion [i] based on digital (49, 170, 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]

(170−107, 170, 99)-Net over F₄ — Digital

Digital (63, 170, 99)-net over F₄, using

t-expansion [i] based on digital (61, 170, 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]

(170−107, 170, 528)-Net in Base 4 — Upper bound on s

There is no (63, 170, 529)-net in base 4, because

1 times m-reduction [i] would yield (63, 169, 529)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 596944 825003 839402 057444 863768 206761 761708 249432 483299 880000 394520 648951 262376 262414 581636 005100 805120 > 4¹⁶⁹ [i]