MinT - Best Known (249−51, 249, s)-Nets in Base 4

Best Known (249−51, 249, s)-Nets in Base 4

(249−51, 249, 1539)-Net over F₄ — Constructive and digital

Digital (198, 249, 1539)-net over F₄, using

6 times m-reduction [i] based on digital (198, 255, 1539)-net over F₄, using
- trace code for nets [i] based on digital (28, 85, 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]

(249−51, 249, 6493)-Net over F₄ — Digital

Digital (198, 249, 6493)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−51, 249, 3183735)-Net in Base 4 — Upper bound on s

There is no (198, 249, 3183736)-net in base 4, because

1 times m-reduction [i] would yield (198, 248, 3183736)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 204588 508604 267662 944427 081770 947237 031765 264922 207209 426273 307761 893465 884463 563474 949734 349677 876464 023896 379554 853338 835948 968855 748446 150836 478636 > 4²⁴⁸ [i]