MinT - Best Known (158, 199, s)-Nets in Base 4

Best Known (158, 199, s)-Nets in Base 4

(158, 199, 1061)-Net over F₄ — Constructive and digital

Digital (158, 199, 1061)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 35, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (123, 164, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 41, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(158, 199, 5219)-Net over F₄ — Digital

Digital (158, 199, 5219)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(158, 199, 2526830)-Net in Base 4 — Upper bound on s

There is no (158, 199, 2526831)-net in base 4, because

1 times m-reduction [i] would yield (158, 198, 2526831)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 161391 246550 987702 974751 280279 414133 646193 051859 696263 809396 701157 886465 099869 909372 291341 107434 079007 046219 439516 420494 > 4¹⁹⁸ [i]