MinT - Best Known (137, 166, s)-Nets in Base 3

Best Known (137, 166, s)-Nets in Base 3

(137, 166, 704)-Net over F₃ — Constructive and digital

Digital (137, 166, 704)-net over F₃, using

3¹ times duplication [i] based on digital (136, 165, 704)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 21, 16)-net over F₃, using
    - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
        an explicitly constructive algebraic function field [i]
  2. digital (115, 144, 688)-net over F₃, using
    - trace code for nets [i] based on digital (7, 36, 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]

(137, 166, 4473)-Net over F₃ — Digital

Digital (137, 166, 4473)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶⁶, 4473, F₃, 29) (dual of [4473, 4307, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁶⁶, 6606, F₃, 29) (dual of [6606, 6440, 30]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(3¹⁶⁴, 6604, F₃, 29) (dual of [6604, 6440, 30]-code), using
    - constructionÂ X applied to C_e(28) âŠ‚ C_e(22) [i] based on
      1. 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]
      2. linear OA(3¹²¹, 6561, F₃, 23) (dual of [6561, 6440, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      3. linear OA(3¹¹, 43, F₃, 5) (dual of [43, 32, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(137, 166, 1269515)-Net in Base 3 — Upper bound on s

There is no (137, 166, 1269516)-net in base 3, because

1 times m-reduction [i] would yield (137, 165, 1269516)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 308975 792222 877097 828570 454031 211575 343331 354936 524462 894871 979708 267770 055817 > 3¹⁶⁵ [i]