MinT - Best Known (170−87, 170, s)-Nets in Base 4

Best Known (170−87, 170, s)-Nets in Base 4

(170−87, 170, 104)-Net over F₄ — Constructive and digital

Digital (83, 170, 104)-net over F₄, using

t-expansion [i] based on digital (73, 170, 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]

(170−87, 170, 129)-Net over F₄ — Digital

Digital (83, 170, 129)-net over F₄, using

t-expansion [i] based on digital (81, 170, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(170−87, 170, 1272)-Net in Base 4 — Upper bound on s

There is no (83, 170, 1273)-net in base 4, because

1 times m-reduction [i] would yield (83, 169, 1273)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 560805 747178 482640 751641 848482 350439 554593 101393 524569 981639 105171 951056 501823 445890 689016 111601 222480 > 4¹⁶⁹ [i]