MinT - Best Known (172−73, 172, s)-Nets in Base 4

Best Known (172−73, 172, s)-Nets in Base 4

(172−73, 172, 130)-Net over F₄ — Constructive and digital

Digital (99, 172, 130)-net over F₄, using

14 times m-reduction [i] based on digital (99, 186, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 93, 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]

(172−73, 172, 212)-Net over F₄ — Digital

Digital (99, 172, 212)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−73, 172, 3417)-Net in Base 4 — Upper bound on s

There is no (99, 172, 3418)-net in base 4, because

1 times m-reduction [i] would yield (99, 171, 3418)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 027065 411178 438651 036105 502277 946351 632791 425103 827829 384056 841826 299577 637067 011050 742409 906397 257060 > 4¹⁷¹ [i]