MinT - Best Known (212−78, 212, s)-Nets in Base 4

Best Known (212−78, 212, s)-Nets in Base 4

(212−78, 212, 147)-Net over F₄ — Constructive and digital

Digital (134, 212, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 44, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (90, 168, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 84, 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]

(212−78, 212, 152)-Net in Base 4 — Constructive

(134, 212, 152)-net in base 4, using

2 times m-reduction [i] based on (134, 214, 152)-net in base 4, using
- trace code for nets [i] based on (27, 107, 76)-net in base 16, using
  - 3 times m-reduction [i] based on (27, 110, 76)-net in base 16, using
    - base change [i] based on digital (5, 88, 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]

(212−78, 212, 409)-Net over F₄ — Digital

Digital (134, 212, 409)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−78, 212, 9585)-Net in Base 4 — Upper bound on s

There is no (134, 212, 9586)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43 488880 313000 584457 910622 513413 063147 945220 643898 907196 836142 596358 438000 082506 684632 674272 155115 429940 953634 067305 877714 432744 > 4²¹² [i]