MinT - Best Known (50−17, 50, s)-Nets in Base 4

Best Known (50−17, 50, s)-Nets in Base 4

(50−17, 50, 130)-Net over F₄ — Constructive and digital

Digital (33, 50, 130)-net over F₄, using

4 times m-reduction [i] based on digital (33, 54, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 27, 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]

(50−17, 50, 188)-Net over F₄ — Digital

Digital (33, 50, 188)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵⁰, 188, F₄, 17) (dual of [188, 138, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁵⁰, 255, F₄, 17) (dual of [255, 205, 18]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 255Â = 4⁴âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]

(50−17, 50, 6105)-Net in Base 4 — Upper bound on s

There is no (33, 50, 6106)-net in base 4, because

1 times m-reduction [i] would yield (33, 49, 6106)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 316963 765874 820779 777947 277743 > 4⁴⁹ [i]