MinT - Best Known (134−34, 134, s)-Nets in Base 3

Best Known (134−34, 134, s)-Nets in Base 3

(134−34, 134, 282)-Net over F₃ — Constructive and digital

Digital (100, 134, 282)-net over F₃, using

1 times m-reduction [i] based on digital (100, 135, 282)-net over F₃, using
- trace code for nets [i] based on digital (10, 45, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(134−34, 134, 588)-Net over F₃ — Digital

Digital (100, 134, 588)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³⁴, 588, F₃, 34) (dual of [588, 454, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³⁴, 745, F₃, 34) (dual of [745, 611, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(30) [i] based on
    1. linear OA(3¹³⁰, 729, F₃, 34) (dual of [729, 599, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(3¹¹⁸, 729, F₃, 31) (dual of [729, 611, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    3. linear OA(3⁴, 16, F₃, 2) (dual of [16, 12, 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]

(134−34, 134, 20673)-Net in Base 3 — Upper bound on s

There is no (100, 134, 20674)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8600 945678 182146 292113 002076 517479 799740 311775 691939 505671 355877 > 3¹³⁴ [i]