MinT - Best Known (170−99, 170, s)-Nets in Base 4

Best Known (170−99, 170, s)-Nets in Base 4

(170−99, 170, 66)-Net over F₄ — Constructive and digital

Digital (71, 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]

(170−99, 170, 105)-Net over F₄ — Digital

Digital (71, 170, 105)-net over F₄, using

t-expansion [i] based on digital (70, 170, 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]

(170−99, 170, 720)-Net in Base 4 — Upper bound on s

There is no (71, 170, 721)-net in base 4, because

1 times m-reduction [i] would yield (71, 169, 721)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 588485 425401 968227 103004 364823 979749 024271 948419 431963 085065 528978 960081 606492 480659 366717 446066 311768 > 4¹⁶⁹ [i]