MinT - Best Known (75, 95, s)-Nets in Base 3

Best Known (75, 95, s)-Nets in Base 3

(75, 95, 464)-Net over F₃ — Constructive and digital

Digital (75, 95, 464)-net over F₃, using

t-expansion [i] based on digital (74, 95, 464)-net over F₃, using
- 1 times m-reduction [i] based on digital (74, 96, 464)-net over F₃, using
  - trace code for nets [i] based on digital (2, 24, 116)-net over F₈₁, using
    - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]

(75, 95, 1155)-Net over F₃ — Digital

Digital (75, 95, 1155)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁹⁵, 1155, F₃, 20) (dual of [1155, 1060, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁹⁵, 2200, F₃, 20) (dual of [2200, 2105, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(16) [i] based on
    1. linear OA(3⁹², 2187, F₃, 20) (dual of [2187, 2095, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. linear OA(3⁷⁸, 2187, F₃, 17) (dual of [2187, 2109, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(75, 95, 77187)-Net in Base 3 — Upper bound on s

There is no (75, 95, 77188)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2121 139628 640788 713438 394945 348466 576892 794297 > 3⁹⁵ [i]