MinT - Best Known (239−84, 239, s)-Nets in Base 4

Best Known (239−84, 239, s)-Nets in Base 4

(239−84, 239, 163)-Net over F₄ — Constructive and digital

Digital (155, 239, 163)-net over F₄, using

t-expansion [i] based on 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]

(239−84, 239, 240)-Net in Base 4 — Constructive

(155, 239, 240)-net in base 4, using

1 times m-reduction [i] based on (155, 240, 240)-net in base 4, using
- trace code for nets [i] based on (35, 120, 120)-net in base 16, using
  - base change [i] based on digital (11, 96, 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]

(239−84, 239, 534)-Net over F₄ — Digital

Digital (155, 239, 534)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−84, 239, 14644)-Net in Base 4 — Upper bound on s

There is no (155, 239, 14645)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 782076 038471 163331 985129 388740 335902 977824 576758 975826 172165 053836 079442 668562 285331 619034 552000 282154 743025 992952 961893 156247 390082 597743 348348 > 4²³⁹ [i]