MinT - Best Known (33−13, 33, s)-Nets in Base 4

Best Known (33−13, 33, s)-Nets in Base 4

(33−13, 33, 76)-Net over F₄ — Constructive and digital

Digital (20, 33, 76)-net over F₄, using

1 times m-reduction [i] based on digital (20, 34, 76)-net over F₄, using
- trace code for nets [i] based on digital (3, 17, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(33−13, 33, 86)-Net over F₄ — Digital

Digital (20, 33, 86)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4³³, 86, F₄, 13) (dual of [86, 53, 14]-code), using
- an extension C_e(12) of the narrow-sense BCH-code C(I) with length 85Â | 4⁴âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(33−13, 33, 1617)-Net in Base 4 — Upper bound on s

There is no (20, 33, 1618)-net in base 4, because

1 times m-reduction [i] would yield (20, 32, 1618)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 18 471205 125119 975320 > 4³² [i]