MinT - Best Known (89, 89+65, s)-Nets in Base 4

Best Known (89, 89+65, s)-Nets in Base 4

(89, 89+65, 130)-Net over F₄ — Constructive and digital

Digital (89, 154, 130)-net over F₄, using

12 times m-reduction [i] based on digital (89, 166, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 83, 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]

(89, 89+65, 196)-Net over F₄ — Digital

Digital (89, 154, 196)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+65, 3197)-Net in Base 4 — Upper bound on s

There is no (89, 154, 3198)-net in base 4, because

1 times m-reduction [i] would yield (89, 153, 3198)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 130 833115 219875 124789 383650 879256 320596 610797 899609 878714 774759 328845 229317 327585 037851 108005 > 4¹⁵³ [i]