MinT - Best Known (165−31, 165, s)-Nets in Base 3

Best Known (165−31, 165, s)-Nets in Base 3

(165−31, 165, 688)-Net over F₃ — Constructive and digital

Digital (134, 165, 688)-net over F₃, using

t-expansion [i] based on digital (133, 165, 688)-net over F₃, using
- 3 times m-reduction [i] based on digital (133, 168, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 42, 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]

(165−31, 165, 3290)-Net over F₃ — Digital

Digital (134, 165, 3290)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁶⁵, 3290, F₃, 2, 31) (dual of [(3290, 2), 6415, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹⁶⁵, 6580, F₃, 31) (dual of [6580, 6415, 32]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹⁶⁵, 6581, F₃, 31) (dual of [6581, 6416, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(27) [i] based on
      1. linear OA(3¹⁶¹, 6561, F₃, 31) (dual of [6561, 6400, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(3¹⁴⁵, 6561, F₃, 28) (dual of [6561, 6416, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      3. linear OA(3⁴, 20, F₃, 2) (dual of [20, 16, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(165−31, 165, 528750)-Net in Base 3 — Upper bound on s

There is no (134, 165, 528751)-net in base 3, because

1 times m-reduction [i] would yield (134, 164, 528751)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 769689 161697 926277 541164 795953 604789 747186 790447 667297 983499 927794 688792 767907 > 3¹⁶⁴ [i]