MinT - Best Known (68, 68+41, s)-Nets in Base 4

Best Known (68, 68+41, s)-Nets in Base 4

(68, 68+41, 130)-Net over F₄ — Constructive and digital

Digital (68, 109, 130)-net over F₄, using

15 times m-reduction [i] based on digital (68, 124, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 62, 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]

(68, 68+41, 216)-Net over F₄ — Digital

Digital (68, 109, 216)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+41, 4919)-Net in Base 4 — Upper bound on s

There is no (68, 109, 4920)-net in base 4, because

1 times m-reduction [i] would yield (68, 108, 4920)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 105696 549084 243763 499172 573188 288735 220961 553060 446394 299679 511469 > 4¹⁰⁸ [i]