MinT - Best Known (96−20, 96, s)-Nets in Base 3

Best Known (96−20, 96, s)-Nets in Base 3

(96−20, 96, 600)-Net over F₃ — Constructive and digital

Digital (76, 96, 600)-net over F₃, using

trace code for nets [i] based on digital (4, 24, 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]

(96−20, 96, 1229)-Net over F₃ — Digital

Digital (76, 96, 1229)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁹⁶, 1229, F₃, 20) (dual of [1229, 1133, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁹⁶, 2205, F₃, 20) (dual of [2205, 2109, 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(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]

(96−20, 96, 86151)-Net in Base 3 — Upper bound on s

There is no (76, 96, 86152)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6363 109724 648574 155755 655367 619885 034792 994609 > 3⁹⁶ [i]