MinT - Best Known (189, 223, s)-Nets in Base 4

Best Known (189, 223, s)-Nets in Base 4

(189, 223, 3872)-Net over F₄ — Constructive and digital

Digital (189, 223, 3872)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 22, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (167, 201, 3855)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²⁰¹, 3855, F₄, 34, 34) (dual of [(3855, 34), 130869, 35]-NRT-code), using
    - OA 17-folding and stacking [i] based on linear OA(4²⁰¹, 65535, F₄, 34) (dual of [65535, 65334, 35]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²⁰¹, 65536, F₄, 34) (dual of [65536, 65335, 35]-code), using
        an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(189, 223, 64029)-Net over F₄ — Digital

Digital (189, 223, 64029)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²²³, 64029, F₄, 34) (dual of [64029, 63806, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4²²³, 65561, F₄, 34) (dual of [65561, 65338, 35]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(4²², 25, F₄, 17) (dual of [25, 3, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²², 26, F₄, 17) (dual of [26, 4, 18]-code), using
        2 times truncation [i] based on linear OA(4²⁴, 28, F₄, 19) (dual of [28, 4, 20]-code), using
        an explicit generator matrix [i]
    2. linear OA(4²⁰¹, 65536, F₄, 34) (dual of [65536, 65335, 35]-code), using
      - an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(189, 223, large)-Net in Base 4 — Upper bound on s

There is no (189, 223, large)-net in base 4, because

32 times m-reduction [i] would yield (189, 191, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]