MinT - Best Known (154, 154+85, s)-Nets in Base 4

Best Known (154, 154+85, s)-Nets in Base 4

(154, 154+85, 163)-Net over F₄ — Constructive and digital

Digital (154, 239, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 57, 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 (97, 182, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 91, 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]

(154, 154+85, 208)-Net in Base 4 — Constructive

(154, 239, 208)-net in base 4, using

t-expansion [i] based on (153, 239, 208)-net in base 4, using
- 1 times m-reduction [i] based on (153, 240, 208)-net in base 4, using
  - trace code for nets [i] based on (33, 120, 104)-net in base 16, using
    - base change [i] based on digital (9, 96, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(154, 154+85, 512)-Net over F₄ — Digital

Digital (154, 239, 512)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(154, 154+85, 14167)-Net in Base 4 — Upper bound on s

There is no (154, 239, 14168)-net in base 4, because

1 times m-reduction [i] would yield (154, 238, 14168)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195296 654676 423823 151808 423769 757181 952745 542899 106983 781285 153553 625395 104080 142426 887338 316301 188263 084735 769627 530146 947557 068050 583185 858412 > 4²³⁸ [i]