MinT - Best Known (157, 186, s)-Nets in Base 3

Best Known (157, 186, s)-Nets in Base 3

(157, 186, 1480)-Net over F₃ — Constructive and digital

Digital (157, 186, 1480)-net over F₃, using

2 times m-reduction [i] based on digital (157, 188, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 47, 370)-net over F₈₁, using
  - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
      - a function field by GarcÃa and Quoos [i]

(157, 186, 10126)-Net over F₃ — Digital

Digital (157, 186, 10126)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸⁶, 10126, F₃, 29) (dual of [10126, 9940, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸⁶, 19698, F₃, 29) (dual of [19698, 19512, 30]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹⁴, 15, F₃, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
      - dual of repetition code with length 15 [i]
    2. linear OA(3¹⁷², 19683, F₃, 29) (dual of [19683, 19511, 30]-code), using
      - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(157, 186, 6098785)-Net in Base 3 — Upper bound on s

There is no (157, 186, 6098786)-net in base 3, because

1 times m-reduction [i] would yield (157, 185, 6098786)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18511 114686 438793 435064 747389 549159 549379 533521 058708 368946 613459 053963 611762 317092 601037 > 3¹⁸⁵ [i]