MinT - Best Known (180−38, 180, s)-Nets in Base 3

Best Known (180−38, 180, s)-Nets in Base 3

(180−38, 180, 688)-Net over F₃ — Constructive and digital

Digital (142, 180, 688)-net over F₃, using

trace code for nets [i] based on digital (7, 45, 172)-net over F₈₁, using
- net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
    - a function field by SÃ©mirat [i]

(180−38, 180, 1650)-Net over F₃ — Digital

Digital (142, 180, 1650)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁸⁰, 1650, F₃, 38) (dual of [1650, 1470, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁸⁰, 2205, F₃, 38) (dual of [2205, 2025, 39]-code), using
  - constructionÂ X applied to C_e(37) âŠ‚ C_e(34) [i] based on
    1. linear OA(3¹⁷⁶, 2187, F₃, 38) (dual of [2187, 2011, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
    2. linear OA(3¹⁶², 2187, F₃, 35) (dual of [2187, 2025, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [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]

(180−38, 180, 131286)-Net in Base 3 — Upper bound on s

There is no (142, 180, 131287)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 76 181701 055665 300804 062318 542106 418404 359579 792728 542964 363871 284356 908691 316146 650427 > 3¹⁸⁰ [i]