MinT - Best Known (103, 131, s)-Nets in Base 3

Best Known (103, 131, s)-Nets in Base 3

(103, 131, 600)-Net over F₃ — Constructive and digital

Digital (103, 131, 600)-net over F₃, using

1 times m-reduction [i] based on digital (103, 132, 600)-net over F₃, using
- trace code for nets [i] based on digital (4, 33, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(103, 131, 1258)-Net over F₃ — Digital

Digital (103, 131, 1258)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³¹, 1258, F₃, 28) (dual of [1258, 1127, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³¹, 2205, F₃, 28) (dual of [2205, 2074, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(24) [i] based on
    1. linear OA(3¹²⁷, 2187, F₃, 28) (dual of [2187, 2060, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(3¹¹³, 2187, F₃, 25) (dual of [2187, 2074, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    3. linear OA(3⁴, 18, F₃, 2) (dual of [18, 14, 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]

(103, 131, 88075)-Net in Base 3 — Upper bound on s

There is no (103, 131, 88076)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 318 382769 533344 878696 405697 171087 658631 021088 560529 918596 280969 > 3¹³¹ [i]