MinT - Best Known (81, 81+63, s)-Nets in Base 4

Best Known (81, 81+63, s)-Nets in Base 4

(81, 81+63, 130)-Net over F₄ — Constructive and digital

Digital (81, 144, 130)-net over F₄, using

6 times m-reduction [i] based on digital (81, 150, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 75, 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]

(81, 81+63, 165)-Net over F₄ — Digital

Digital (81, 144, 165)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+63, 2453)-Net in Base 4 — Upper bound on s

There is no (81, 144, 2454)-net in base 4, because

1 times m-reduction [i] would yield (81, 143, 2454)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 125 771507 770682 014959 815073 581814 144376 085245 722800 518124 253287 360466 046222 054807 493056 > 4¹⁴³ [i]