MinT - Best Known (83−23, 83, s)-Nets in Base 4

Best Known (83−23, 83, s)-Nets in Base 4

(83−23, 83, 312)-Net over F₄ — Constructive and digital

Digital (60, 83, 312)-net over F₄, using

t-expansion [i] based on digital (59, 83, 312)-net over F₄, using
- 1 times m-reduction [i] based on digital (59, 84, 312)-net over F₄, using
  - trace code for nets [i] based on digital (3, 28, 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]

(83−23, 83, 387)-Net in Base 4 — Constructive

(60, 83, 387)-net in base 4, using

1 times m-reduction [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]

(83−23, 83, 575)-Net over F₄ — Digital

Digital (60, 83, 575)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(83−23, 83, 50338)-Net in Base 4 — Upper bound on s

There is no (60, 83, 50339)-net in base 4, because

1 times m-reduction [i] would yield (60, 82, 50339)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 23 385726 107207 625176 316471 437235 609262 876422 917104 > 4⁸² [i]