MinT - Best Known (178−60, 178, s)-Nets in Base 4

Best Known (178−60, 178, s)-Nets in Base 4

(178−60, 178, 163)-Net over F₄ — Constructive and digital

Digital (118, 178, 163)-net over F₄, using

1 times m-reduction [i] based on digital (118, 179, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 45, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 134, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 67, 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]

(178−60, 178, 240)-Net in Base 4 — Constructive

(118, 178, 240)-net in base 4, using

trace code for nets [i] based on (29, 89, 120)-net in base 16, using
- 1 times m-reduction [i] based on (29, 90, 120)-net in base 16, using
  - base change [i] based on digital (11, 72, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(178−60, 178, 477)-Net over F₄ — Digital

Digital (118, 178, 477)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−60, 178, 14968)-Net in Base 4 — Upper bound on s

There is no (118, 178, 14969)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 146911 205442 028669 597225 865577 638999 585252 239874 864490 717300 274842 005492 010474 079953 999036 451674 297871 959664 > 4¹⁷⁸ [i]