MinT - Best Known (191−33, 191, s)-Nets in Base 3

Best Known (191−33, 191, s)-Nets in Base 3

(191−33, 191, 896)-Net over F₃ — Constructive and digital

Digital (158, 191, 896)-net over F₃, using

t-expansion [i] based on digital (157, 191, 896)-net over F₃, using
- 1 times m-reduction [i] based on digital (157, 192, 896)-net over F₃, using
  - trace code for nets [i] based on digital (13, 48, 224)-net over F₈₁, using
    - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
        a function field by SÃ©mirat [i]

(191−33, 191, 5186)-Net over F₃ — Digital

Digital (158, 191, 5186)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹¹, 5186, F₃, 33) (dual of [5186, 4995, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹¹, 6614, F₃, 33) (dual of [6614, 6423, 34]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(25) [i] based on
    1. linear OA(3¹⁷⁷, 6561, F₃, 34) (dual of [6561, 6384, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(3¹³⁷, 6561, F₃, 26) (dual of [6561, 6424, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(3¹⁴, 53, F₃, 6) (dual of [53, 39, 7]-code), using
      - codes constructed by inverting constructionÂ Y₁ [i]

(191−33, 191, 1575140)-Net in Base 3 — Upper bound on s

There is no (158, 191, 1575141)-net in base 3, because

1 times m-reduction [i] would yield (158, 190, 1575141)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 498209 635911 128214 196074 167928 225225 236431 781597 352367 108932 311433 029027 226213 492351 477185 > 3¹⁹⁰ [i]