MinT - Best Known (137, 137+77, s)-Nets in Base 4

Best Known (137, 137+77, s)-Nets in Base 4

(137, 137+77, 157)-Net over F₄ — Constructive and digital

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]

(137, 137+77, 196)-Net in Base 4 — Constructive

(137, 214, 196)-net in base 4, using

2 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]

(137, 137+77, 446)-Net over F₄ — Digital

Digital (137, 214, 446)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+77, 11837)-Net in Base 4 — Upper bound on s

There is no (137, 214, 11838)-net in base 4, because

1 times m-reduction [i] would yield (137, 213, 11838)-net in base 4, but
- 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]