MinT - Best Known (221−43, 221, s)-Nets in Base 4

Best Known (221−43, 221, s)-Nets in Base 4

(221−43, 221, 1539)-Net over F₄ — Constructive and digital

Digital (178, 221, 1539)-net over F₄, using

4 times m-reduction [i] based on digital (178, 225, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 75, 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]

(221−43, 221, 8124)-Net over F₄ — Digital

Digital (178, 221, 8124)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(221−43, 221, 5870366)-Net in Base 4 — Upper bound on s

There is no (178, 221, 5870367)-net in base 4, because

1 times m-reduction [i] would yield (178, 220, 5870367)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 839220 765375 007458 224824 152710 068985 973235 252677 592431 338732 324913 490190 137807 473818 699248 867737 273104 853684 097982 125821 633183 889718 > 4²²⁰ [i]