MinT - Best Known (159, 159+40, s)-Nets in Base 4

Best Known (159, 159+40, s)-Nets in Base 4

(159, 159+40, 1065)-Net over F₄ — Constructive and digital

Digital (159, 199, 1065)-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 (124, 164, 1032)-net over F₄, using
  - trace code for nets [i] based on digital (1, 41, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(159, 159+40, 6078)-Net over F₄ — Digital

Digital (159, 199, 6078)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(159, 159+40, 2708191)-Net in Base 4 — Upper bound on s

There is no (159, 199, 2708192)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 645567 034092 274060 152632 835891 558885 009348 110489 564010 670148 408023 659438 025312 361998 781650 198623 418524 765137 305926 628519 > 4¹⁹⁹ [i]