MinT - Best Known (125, 125+57, s)-Nets in Base 4

Best Known (125, 125+57, s)-Nets in Base 4

(125, 125+57, 312)-Net over F₄ — Constructive and digital

Digital (125, 182, 312)-net over F₄, using

1 times m-reduction [i] based on digital (125, 183, 312)-net over F₄, using
- trace code for nets [i] based on digital (3, 61, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(125, 125+57, 644)-Net over F₄ — Digital

Digital (125, 182, 644)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125, 125+57, 29338)-Net in Base 4 — Upper bound on s

There is no (125, 182, 29339)-net in base 4, because

1 times m-reduction [i] would yield (125, 181, 29339)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 401133 444981 441839 579324 080870 226245 751428 898196 038444 970228 405387 049377 994912 492699 055670 549514 128018 902656 > 4¹⁸¹ [i]