MinT - Best Known (212−79, 212, s)-Nets in Base 4

Best Known (212−79, 212, s)-Nets in Base 4

(212−79, 212, 144)-Net over F₄ — Constructive and digital

Digital (133, 212, 144)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 42, 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 (91, 170, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 85, 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]

(212−79, 212, 152)-Net in Base 4 — Constructive

(133, 212, 152)-net in base 4, using

4² times duplication [i] based on (131, 210, 152)-net in base 4, using
- trace code for nets [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]

(212−79, 212, 391)-Net over F₄ — Digital

Digital (133, 212, 391)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−79, 212, 9249)-Net in Base 4 — Upper bound on s

There is no (133, 212, 9250)-net in base 4, because

1 times m-reduction [i] would yield (133, 211, 9250)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 866452 945711 517855 011545 521640 993595 574370 732469 957182 061809 451596 296715 703094 882959 615369 833887 548392 690778 997200 901090 535936 > 4²¹¹ [i]