MinT - Best Known (239−83, 239, s)-Nets in Base 4

Best Known (239−83, 239, s)-Nets in Base 4

(239−83, 239, 163)-Net over F₄ — Constructive and digital

Digital (156, 239, 163)-net over F₄, using

3 times m-reduction [i] based on digital (156, 242, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 58, 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 (98, 184, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 92, 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]

(239−83, 239, 240)-Net in Base 4 — Constructive

(156, 239, 240)-net in base 4, using

t-expansion [i] based on (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]

(239−83, 239, 556)-Net over F₄ — Digital

Digital (156, 239, 556)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−83, 239, 16780)-Net in Base 4 — Upper bound on s

There is no (156, 239, 16781)-net in base 4, because

1 times m-reduction [i] would yield (156, 238, 16781)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195314 255130 659589 641851 434433 782514 852515 086616 752460 700963 136578 207286 825055 341601 202933 298711 877175 062053 012944 328377 637415 813527 615842 007104 > 4²³⁸ [i]