MinT - Best Known (186−19, 186, s)-Nets in Base 4

Best Known (186−19, 186, s)-Nets in Base 4

(186−19, 186, 932090)-Net over F₄ — Constructive and digital

Digital (167, 186, 932090)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (8, 17, 24)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 5, 10)-net over F₄, using
      - linear code embedding found in a computer search [i]
    2. digital (3, 12, 14)-net over F₄, using
      - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
        an explicitly constructive algebraic function field [i]
2. digital (150, 169, 932066)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁶⁹, 932066, F₄, 19, 19) (dual of [(932066, 19), 17709085, 20]-NRT-code), using
    - OOA 9-folding and stacking with additional row [i] based on linear OA(4¹⁶⁹, 8388595, F₄, 19) (dual of [8388595, 8388426, 20]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁶⁹, large, F₄, 19) (dual of [large, large−169, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(186−19, 186, large)-Net over F₄ — Digital

Digital (167, 186, large)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁶, large, F₄, 19) (dual of [large, large−186, 20]-code), using
- 17 times code embedding in larger space [i] based on linear OA(4¹⁶⁹, large, F₄, 19) (dual of [large, large−169, 20]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(186−19, 186, large)-Net in Base 4 — Upper bound on s

There is no (167, 186, large)-net in base 4, because

17 times m-reduction [i] would yield (167, 169, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]