MinT - Best Known (164−63, 164, s)-Nets in Base 4

Best Known (164−63, 164, s)-Nets in Base 4

(164−63, 164, 130)-Net over F₄ — Constructive and digital

Digital (101, 164, 130)-net over F₄, using

26 times m-reduction [i] based on digital (101, 190, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 95, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(164−63, 164, 282)-Net over F₄ — Digital

Digital (101, 164, 282)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−63, 164, 6036)-Net in Base 4 — Upper bound on s

There is no (101, 164, 6037)-net in base 4, because

1 times m-reduction [i] would yield (101, 163, 6037)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 137 240136 511870 592230 464139 559137 424858 567640 518143 111310 836857 227798 600004 821882 162097 891931 665280 > 4¹⁶³ [i]