MinT - Best Known (213−76, 213, s)-Nets in Base 4

Best Known (213−76, 213, s)-Nets in Base 4

(213−76, 213, 157)-Net over F₄ — Constructive and digital

Digital (137, 213, 157)-net over F₄, using

1 times m-reduction [i] based on digital (137, 214, 157)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 48, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, 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]

(213−76, 213, 196)-Net in Base 4 — Constructive

(137, 213, 196)-net in base 4, using

3 times m-reduction [i] based on (137, 216, 196)-net in base 4, using
- trace code for nets [i] based on (29, 108, 98)-net in base 16, using
  - 2 times m-reduction [i] based on (29, 110, 98)-net in base 16, using
    - base change [i] based on digital (7, 88, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(213−76, 213, 457)-Net over F₄ — Digital

Digital (137, 213, 457)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−76, 213, 11837)-Net in Base 4 — Upper bound on s

There is no (137, 213, 11838)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 173 785004 318958 478073 519460 178595 235240 676011 002585 976341 614189 688415 255257 120577 082158 438422 732063 197391 168361 367698 621331 749340 > 4²¹³ [i]