MinT - Best Known (89, 89+162, s)-Nets in Base 4

Best Known (89, 89+162, s)-Nets in Base 4

(89, 89+162, 104)-Net over F₄ — Constructive and digital

Digital (89, 251, 104)-net over F₄, using

t-expansion [i] based on digital (73, 251, 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+162, 129)-Net over F₄ — Digital

Digital (89, 251, 129)-net over F₄, using

t-expansion [i] based on digital (81, 251, 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+162, 692)-Net in Base 4 — Upper bound on s

There is no (89, 251, 693)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 471562 468196 189463 571960 174224 100374 926877 907002 329782 568631 335498 739343 513851 905959 835803 499327 316029 416682 938429 822636 419396 492259 305800 427339 789600 > 4²⁵¹ [i]