MinT - Best Known (119, 119+32, s)-Nets in Base 3

Best Known (119, 119+32, s)-Nets in Base 3

(119, 119+32, 640)-Net over F₃ — Constructive and digital

Digital (119, 151, 640)-net over F₃, using

1 times m-reduction [i] based on digital (119, 152, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 38, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(119, 119+32, 1436)-Net over F₃ — Digital

Digital (119, 151, 1436)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵¹, 1436, F₃, 32) (dual of [1436, 1285, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁵¹, 2200, F₃, 32) (dual of [2200, 2049, 33]-code), using
  - constructionÂ X applied to C_e(31) âŠ‚ C_e(28) [i] based on
    1. linear OA(3¹⁴⁸, 2187, F₃, 32) (dual of [2187, 2039, 33]-code), using an extension C_e(31) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,31], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
    2. linear OA(3¹³⁴, 2187, F₃, 29) (dual of [2187, 2053, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(119, 119+32, 108212)-Net in Base 3 — Upper bound on s

There is no (119, 151, 108213)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 110009 017296 671973 375160 342753 999581 944816 670824 778846 942017 216217 075137 > 3¹⁵¹ [i]