MinT - Best Known (130, 130+77, s)-Nets in Base 4

Best Known (130, 130+77, s)-Nets in Base 4

(130, 130+77, 144)-Net over F₄ — Constructive and digital

Digital (130, 207, 144)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 41, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (89, 166, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 83, 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]

(130, 130+77, 152)-Net in Base 4 — Constructive

(130, 207, 152)-net in base 4, using

1 times m-reduction [i] based on (130, 208, 152)-net in base 4, using
- trace code for nets [i] based on (26, 104, 76)-net in base 16, using
  - 1 times m-reduction [i] based on (26, 105, 76)-net in base 16, using
    - base change [i] based on digital (5, 84, 76)-net over F₃₂, using
      - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(130, 130+77, 385)-Net over F₄ — Digital

Digital (130, 207, 385)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(130, 130+77, 9162)-Net in Base 4 — Upper bound on s

There is no (130, 207, 9163)-net in base 4, because

1 times m-reduction [i] would yield (130, 206, 9163)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10604 366916 792322 876723 529998 791136 982435 724386 424796 101884 941633 350575 170851 040989 337700 027657 381008 533423 580643 909462 486260 > 4²⁰⁶ [i]