MinT - Best Known (45−16, 45, s)-Nets in Base 4

Best Known (45−16, 45, s)-Nets in Base 4

(45−16, 45, 130)-Net over F₄ — Constructive and digital

Digital (29, 45, 130)-net over F₄, using

1 times m-reduction [i] based on digital (29, 46, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 23, 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]

(45−16, 45, 145)-Net over F₄ — Digital

Digital (29, 45, 145)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴⁵, 145, F₄, 16) (dual of [145, 100, 17]-code), using
- 50 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 6 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0) [i] based on linear OA(4³⁵, 85, F₄, 16) (dual of [85, 50, 17]-code), using
  - a â€œGraCycâ€ code from Grasslâ€™s database [i]

(45−16, 45, 3049)-Net in Base 4 — Upper bound on s

There is no (29, 45, 3050)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1238 397853 630793 477821 905181 > 4⁴⁵ [i]