MinT - Best Known (164−61, 164, s)-Nets in Base 4

Best Known (164−61, 164, s)-Nets in Base 4

(164−61, 164, 135)-Net over F₄ — Constructive and digital

Digital (103, 164, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 30, 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 (73, 134, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 67, 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]

(164−61, 164, 316)-Net over F₄ — Digital

Digital (103, 164, 316)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−61, 164, 7472)-Net in Base 4 — Upper bound on s

There is no (103, 164, 7473)-net in base 4, because

1 times m-reduction [i] would yield (103, 163, 7473)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 137 220311 063269 784431 295272 135798 758830 131491 646543 585159 157807 877644 101462 727142 726549 790944 437008 > 4¹⁶³ [i]