MinT - Best Known (122−44, 122, s)-Nets in Base 4

Best Known (122−44, 122, s)-Nets in Base 4

(122−44, 122, 135)-Net over F₄ — Constructive and digital

Digital (78, 122, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 22, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (56, 100, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 50, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(122−44, 122, 273)-Net over F₄ — Digital

Digital (78, 122, 273)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(122−44, 122, 6565)-Net in Base 4 — Upper bound on s

There is no (78, 122, 6566)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 28 340545 684277 129478 895323 665078 311791 072291 133815 973759 076619 260902 285024 > 4¹²² [i]