MinT - Best Known (174−65, 174, s)-Nets in Base 4

Best Known (174−65, 174, s)-Nets in Base 4

(174−65, 174, 135)-Net over F₄ — Constructive and digital

Digital (109, 174, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 32, 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 (77, 142, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 71, 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]

(174−65, 174, 327)-Net over F₄ — Digital

Digital (109, 174, 327)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(174−65, 174, 7641)-Net in Base 4 — Upper bound on s

There is no (109, 174, 7642)-net in base 4, because

1 times m-reduction [i] would yield (109, 173, 7642)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 143 805579 741981 103677 806230 583132 298348 032894 322448 523732 803626 316876 776182 899908 061973 009451 686220 672550 > 4¹⁷³ [i]