MinT - Best Known (156, 241, s)-Nets in Base 4

Best Known (156, 241, s)-Nets in Base 4

(156, 241, 163)-Net over F₄ — Constructive and digital

Digital (156, 241, 163)-net over F₄, using

1 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]

(156, 241, 240)-Net in Base 4 — Constructive

(156, 241, 240)-net in base 4, using

4¹ times duplication [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]

(156, 241, 531)-Net over F₄ — Digital

Digital (156, 241, 531)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(156, 241, 15136)-Net in Base 4 — Upper bound on s

There is no (156, 241, 15137)-net in base 4, because

1 times m-reduction [i] would yield (156, 240, 15137)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 123030 898818 880827 922488 187146 842965 419631 000447 165594 696893 845217 490273 382137 207552 575506 760056 044711 536504 492773 062035 040300 235635 813814 715520 > 4²⁴⁰ [i]