MinT - Best Known (82, 82+47, s)-Nets in Base 4

Best Known (82, 82+47, s)-Nets in Base 4

(82, 82+47, 135)-Net over F₄ — Constructive and digital

Digital (82, 129, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 23, 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 (59, 106, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 53, 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]

(82, 82+47, 276)-Net over F₄ — Digital

Digital (82, 129, 276)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+47, 7027)-Net in Base 4 — Upper bound on s

There is no (82, 129, 7028)-net in base 4, because

1 times m-reduction [i] would yield (82, 128, 7028)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 116116 295547 380347 249047 663113 691705 539740 058900 546186 775938 945267 772765 372280 > 4¹²⁸ [i]