MinT - Best Known (95, 120, s)-Nets in Base 3

Best Known (95, 120, s)-Nets in Base 3

(95, 120, 640)-Net over F₃ — Constructive and digital

Digital (95, 120, 640)-net over F₃, using

trace code for nets [i] based on digital (5, 30, 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]

(95, 120, 1366)-Net over F₃ — Digital

Digital (95, 120, 1366)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹²⁰, 1366, F₃, 25) (dual of [1366, 1246, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹²⁰, 2214, F₃, 25) (dual of [2214, 2094, 26]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(19) [i] based on
    1. 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]
    2. 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]
    3. linear OA(3⁷, 27, F₃, 4) (dual of [27, 20, 5]-code), using
      - an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 26Â = 3³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]

(95, 120, 142478)-Net in Base 3 — Upper bound on s

There is no (95, 120, 142479)-net in base 3, because

1 times m-reduction [i] would yield (95, 119, 142479)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 599 037340 493489 215128 039627 757875 366178 068573 101792 929513 > 3¹¹⁹ [i]