MinT - Best Known (152, 152+41, s)-Nets in Base 3

Best Known (152, 152+41, s)-Nets in Base 3

(152, 152+41, 688)-Net over F₃ — Constructive and digital

Digital (152, 193, 688)-net over F₃, using

3¹ times duplication [i] based on digital (151, 192, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 48, 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]

(152, 152+41, 1683)-Net over F₃ — Digital

Digital (152, 193, 1683)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹³, 1683, F₃, 41) (dual of [1683, 1490, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹³, 2200, F₃, 41) (dual of [2200, 2007, 42]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(37) [i] based on
    1. linear OA(3¹⁹⁰, 2187, F₃, 41) (dual of [2187, 1997, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
    2. 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]
    3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
      - Hamming code H(3,3) [i]

(152, 152+41, 157974)-Net in Base 3 — Upper bound on s

There is no (152, 193, 157975)-net in base 3, because

1 times m-reduction [i] would yield (152, 192, 157975)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 40 486908 665464 309812 549532 037902 441090 325240 245929 269238 801896 959953 130531 644027 886281 532761 > 3¹⁹² [i]