MinT - Best Known (127, 127+45, s)-Nets in Base 4

Best Known (127, 127+45, s)-Nets in Base 4

(127, 127+45, 531)-Net over F₄ — Constructive and digital

Digital (127, 172, 531)-net over F₄, using

8 times m-reduction [i] based on digital (127, 180, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 60, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(127, 127+45, 576)-Net in Base 4 — Constructive

(127, 172, 576)-net in base 4, using

4⁴ times duplication [i] based on (123, 168, 576)-net in base 4, using
- trace code for nets [i] based on (11, 56, 192)-net in base 64, using
  - base change [i] based on digital (3, 48, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(127, 127+45, 1320)-Net over F₄ — Digital

Digital (127, 172, 1320)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+45, 144319)-Net in Base 4 — Upper bound on s

There is no (127, 172, 144320)-net in base 4, because

1 times m-reduction [i] would yield (127, 171, 144320)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 959111 761597 719079 259181 316030 166271 559558 254415 652927 301678 651494 341358 531508 125958 363560 064422 582909 > 4¹⁷¹ [i]