MinT - Best Known (227−45, 227, s)-Nets in Base 4

Best Known (227−45, 227, s)-Nets in Base 4

(227−45, 227, 1539)-Net over F₄ — Constructive and digital

Digital (182, 227, 1539)-net over F₄, using

4 times m-reduction [i] based on digital (182, 231, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 77, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(227−45, 227, 7365)-Net over F₄ — Digital

Digital (182, 227, 7365)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−45, 227, 4618789)-Net in Base 4 — Upper bound on s

There is no (182, 227, 4618790)-net in base 4, because

1 times m-reduction [i] would yield (182, 226, 4618790)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11629 436523 556528 543303 362129 494608 994504 833527 592145 751520 827846 787650 727691 798349 708221 913618 284618 635527 887550 952493 481311 438687 808736 > 4²²⁶ [i]