MinT - Best Known (186, 186+21, s)-Nets in Base 4

Best Known (186, 186+21, s)-Nets in Base 4

(186, 186+21, 838936)-Net over F₄ — Constructive and digital

Digital (186, 207, 838936)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (16, 26, 76)-net over F₄, using
  - trace code for nets [i] based on digital (3, 13, 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]
2. digital (160, 181, 838860)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁸¹, 838860, F₄, 21, 21) (dual of [(838860, 21), 17615879, 22]-NRT-code), using
    - OOA 10-folding and stacking with additional row [i] based on linear OA(4¹⁸¹, 8388601, F₄, 21) (dual of [8388601, 8388420, 22]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁸¹, large, F₄, 21) (dual of [large, large−181, 22]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(186, 186+21, large)-Net over F₄ — Digital

Digital (186, 207, large)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁰⁷, large, F₄, 21) (dual of [large, large−207, 22]-code), using
- 26 times code embedding in larger space [i] based on linear OA(4¹⁸¹, large, F₄, 21) (dual of [large, large−181, 22]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]

(186, 186+21, large)-Net in Base 4 — Upper bound on s

There is no (186, 207, large)-net in base 4, because

19 times m-reduction [i] would yield (186, 188, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]