MinT - Best Known (68, 68+27, s)-Nets in Base 4

Best Known (68, 68+27, s)-Nets in Base 4

(68, 68+27, 312)-Net over F₄ — Constructive and digital

Digital (68, 95, 312)-net over F₄, using

t-expansion [i] based on digital (67, 95, 312)-net over F₄, using
- 1 times m-reduction [i] based on digital (67, 96, 312)-net over F₄, using
  - trace code for nets [i] based on digital (3, 32, 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]

(68, 68+27, 570)-Net over F₄ — Digital

Digital (68, 95, 570)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+27, 42612)-Net in Base 4 — Upper bound on s

There is no (68, 95, 42613)-net in base 4, because

1 times m-reduction [i] would yield (68, 94, 42613)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 392 384823 383183 877333 597853 741556 022052 787612 117592 805480 > 4⁹⁴ [i]