MinT - Best Known (172−32, 172, s)-Nets in Base 3

Best Known (172−32, 172, s)-Nets in Base 3

(172−32, 172, 692)-Net over F₃ — Constructive and digital

Digital (140, 172, 692)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (0, 16, 4)-net over F₃, using
  - net from sequence [i] based on digital (0, 3)-sequence over F₃, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 0 and N(F) ≥ 4, using
      - the rational function field F₃(x) [i]
    - Niederreiter sequence [i]
2. digital (124, 156, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 39, 172)-net over F₈₁, using
    - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
        a function field by SÃ©mirat [i]

(172−32, 172, 3287)-Net over F₃ — Digital

Digital (140, 172, 3287)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷², 3287, F₃, 2, 32) (dual of [(3287, 2), 6402, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹⁷², 6574, F₃, 32) (dual of [6574, 6402, 33]-code), using
  - constructionÂ X applied to C_e(31) âŠ‚ C_e(28) [i] based on
    1. linear OA(3¹⁶⁹, 6561, F₃, 32) (dual of [6561, 6392, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    2. linear OA(3¹⁵³, 6561, F₃, 29) (dual of [6561, 6408, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(172−32, 172, 457664)-Net in Base 3 — Upper bound on s

There is no (140, 172, 457665)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11610 712753 923008 584797 083102 496799 700894 118546 119328 899580 728214 231722 352404 252481 > 3¹⁷² [i]