MinT - Best Known (168, 168+41, s)-Nets in Base 3

Best Known (168, 168+41, s)-Nets in Base 3

(168, 168+41, 688)-Net over F₃ — Constructive and digital

Digital (168, 209, 688)-net over F₃, using

t-expansion [i] based on digital (166, 209, 688)-net over F₃, using
- 3 times m-reduction [i] based on digital (166, 212, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 53, 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]

(168, 168+41, 2491)-Net over F₃ — Digital

Digital (168, 209, 2491)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁰⁹, 2491, F₃, 41) (dual of [2491, 2282, 42]-code), using
- 278 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 11 times 0, 1, 15 times 0, 1, 20 times 0, 1, 26 times 0, 1, 32 times 0, 1, 38 times 0, 1, 46 times 0, 1, 52 times 0) [i] based on linear OA(3¹⁹⁰, 2194, F₃, 41) (dual of [2194, 2004, 42]-code), using
  - constructionÂ X applied to C_e(40) âŠ‚ C_e(39) [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(3¹⁸³, 2187, F₃, 40) (dual of [2187, 2004, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
    3. linear OA(3⁰, 7, F₃, 0) (dual of [7, 7, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(168, 168+41, 380465)-Net in Base 3 — Upper bound on s

There is no (168, 209, 380466)-net in base 3, because

1 times m-reduction [i] would yield (168, 208, 380466)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1742 759677 106197 609981 585810 706371 598960 697638 631791 052729 820766 508725 808700 355969 736741 488423 477193 > 3²⁰⁸ [i]