MinT - Best Known (164−33, 164, s)-Nets in Base 3

Best Known (164−33, 164, s)-Nets in Base 3

(164−33, 164, 688)-Net over F₃ — Constructive and digital

Digital (131, 164, 688)-net over F₃, using

t-expansion [i] based on digital (130, 164, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 41, 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]

(164−33, 164, 1974)-Net over F₃ — Digital

Digital (131, 164, 1974)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶⁴, 1974, F₃, 33) (dual of [1974, 1810, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁶⁴, 2224, F₃, 33) (dual of [2224, 2060, 34]-code), using
  - 1 times truncation [i] based on linear OA(3¹⁶⁵, 2225, F₃, 34) (dual of [2225, 2060, 35]-code), using
    - constructionÂ X applied to C_e(33) âŠ‚ C_e(27) [i] based on
      1. linear OA(3¹⁵⁵, 2187, F₃, 34) (dual of [2187, 2032, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      2. linear OA(3¹²⁷, 2187, F₃, 28) (dual of [2187, 2060, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
      3. linear OA(3¹⁰, 38, F₃, 5) (dual of [38, 28, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹⁰, 39, F₃, 5) (dual of [39, 29, 6]-code), using
        generator matrices provided on Edelâ€™s homepage [i]

(164−33, 164, 246691)-Net in Base 3 — Upper bound on s

There is no (131, 164, 246692)-net in base 3, because

1 times m-reduction [i] would yield (131, 163, 246692)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 589899 412757 254699 829631 187928 325143 566503 949081 670322 338437 855996 892075 222145 > 3¹⁶³ [i]