MinT - Best Known (86−24, 86, s)-Nets in Base 4

Best Known (86−24, 86, s)-Nets in Base 4

(86−24, 86, 312)-Net over F₄ — Constructive and digital

Digital (62, 86, 312)-net over F₄, using

t-expansion [i] based on digital (61, 86, 312)-net over F₄, using
- 1 times m-reduction [i] based on digital (61, 87, 312)-net over F₄, using
  - trace code for nets [i] based on digital (3, 29, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(86−24, 86, 387)-Net in Base 4 — Constructive

(62, 86, 387)-net in base 4, using

4² times duplication [i] based on (60, 84, 387)-net in base 4, using
- trace code for nets [i] based on (4, 28, 129)-net in base 64, using
  - base change [i] based on digital (0, 24, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(86−24, 86, 572)-Net over F₄ — Digital

Digital (62, 86, 572)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−24, 86, 36382)-Net in Base 4 — Upper bound on s

There is no (62, 86, 36383)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5987 708395 356881 274262 470294 349517 758476 083315 324665 > 4⁸⁶ [i]