MinT - Best Known (137−47, 137, s)-Nets in Base 4

Best Known (137−47, 137, s)-Nets in Base 4

(137−47, 137, 151)-Net over F₄ — Constructive and digital

Digital (90, 137, 151)-net over F₄, using

4¹ times duplication [i] based on digital (89, 136, 151)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 30, 21)-net over F₄, using
    - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (59, 106, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 53, 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]

(137−47, 137, 196)-Net in Base 4 — Constructive

(90, 137, 196)-net in base 4, using

1 times m-reduction [i] based on (90, 138, 196)-net in base 4, using
- trace code for nets [i] based on (21, 69, 98)-net in base 16, using
  - 1 times m-reduction [i] based on (21, 70, 98)-net in base 16, using
    - base change [i] based on digital (7, 56, 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]

(137−47, 137, 359)-Net over F₄ — Digital

Digital (90, 137, 359)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−47, 137, 11392)-Net in Base 4 — Upper bound on s

There is no (90, 137, 11393)-net in base 4, because

1 times m-reduction [i] would yield (90, 136, 11393)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7591 243262 562989 647153 260595 295223 014938 849317 617427 226944 493913 022485 039161 993600 > 4¹³⁶ [i]