MinT - Best Known (226−45, 226, s)-Nets in Base 4

Best Known (226−45, 226, s)-Nets in Base 4

(226−45, 226, 1539)-Net over F₄ — Constructive and digital

Digital (181, 226, 1539)-net over F₄, using

t-expansion [i] based on digital (180, 226, 1539)-net over F₄, using
- 2 times m-reduction [i] based on digital (180, 228, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 76, 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]

(226−45, 226, 7137)-Net over F₄ — Digital

Digital (181, 226, 7137)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−45, 226, 4336723)-Net in Base 4 — Upper bound on s

There is no (181, 226, 4336724)-net in base 4, because

1 times m-reduction [i] would yield (181, 225, 4336724)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2907 364856 495408 082729 232502 133015 015110 735682 183666 137578 623851 257209 180604 025682 475913 696497 128079 220448 699253 079646 169096 091673 579760 > 4²²⁵ [i]