MinT - Best Known (89, 89+166, s)-Nets in Base 4

Best Known (89, 89+166, s)-Nets in Base 4

(89, 89+166, 104)-Net over F₄ — Constructive and digital

Digital (89, 255, 104)-net over F₄, using

t-expansion [i] based on digital (73, 255, 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]

(89, 89+166, 129)-Net over F₄ — Digital

Digital (89, 255, 129)-net over F₄, using

t-expansion [i] based on digital (81, 255, 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]

(89, 89+166, 681)-Net in Base 4 — Upper bound on s

There is no (89, 255, 682)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3604 173930 130731 965139 418982 993535 969832 133331 957305 074352 652003 211191 563268 825013 346497 466908 611626 123383 060708 118671 974010 702813 914066 584470 900107 197680 > 4²⁵⁵ [i]