MinT - Best Known (176−18, 176, s)-Nets in Base 4

Best Known (176−18, 176, s)-Nets in Base 4

(176−18, 176, 932101)-Net over F₄ — Constructive and digital

Digital (158, 176, 932101)-net over F₄, using

4¹ times duplication [i] based on digital (157, 175, 932101)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 18, 34)-net over F₄, using
    - trace code for nets [i] based on digital (0, 9, 17)-net over F₁₆, using
      - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
        the rational function field F₁₆(x) [i]
        Niederreiter sequence [i]
  2. digital (139, 157, 932067)-net over F₄, using
    - net defined by OOA [i] based on linear OOA(4¹⁵⁷, 932067, F₄, 18, 18) (dual of [(932067, 18), 16777049, 19]-NRT-code), using
      - OA 9-folding and stacking [i] based on linear OA(4¹⁵⁷, large, F₄, 18) (dual of [large, large−157, 19]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(176−18, 176, large)-Net over F₄ — Digital

Digital (158, 176, large)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷⁶, large, F₄, 18) (dual of [large, large−176, 19]-code), using
- 19 times code embedding in larger space [i] based on linear OA(4¹⁵⁷, large, F₄, 18) (dual of [large, large−157, 19]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 4¹²âˆ’1, defining interval IÂ =Â [0,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]

(176−18, 176, large)-Net in Base 4 — Upper bound on s

There is no (158, 176, large)-net in base 4, because

16 times m-reduction [i] would yield (158, 160, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]