MinT - Best Known (195, 217, s)-Nets in Base 4

Best Known (195, 217, s)-Nets in Base 4

(195, 217, 762648)-Net over F₄ — Constructive and digital

Digital (195, 217, 762648)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 24, 48)-net over F₄, using
  - trace code for nets [i] based on digital (1, 12, 24)-net over F₁₆, using
    - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
2. digital (171, 193, 762600)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁹³, 762600, F₄, 22, 22) (dual of [(762600, 22), 16777007, 23]-NRT-code), using
    - OA 11-folding and stacking [i] based on linear OA(4¹⁹³, 8388600, F₄, 22) (dual of [8388600, 8388407, 23]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁹³, large, F₄, 22) (dual of [large, large−193, 23]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(195, 217, large)-Net over F₄ — Digital

Digital (195, 217, large)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²¹⁷, large, F₄, 22) (dual of [large, large−217, 23]-code), using
- strength reduction [i] based on linear OA(4²¹⁷, large, F₄, 25) (dual of [large, large−217, 26]-code), using
  - the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 4²⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(195, 217, large)-Net in Base 4 — Upper bound on s

There is no (195, 217, large)-net in base 4, because

20 times m-reduction [i] would yield (195, 197, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]