MinT - Best Known (168−35, 168, s)-Nets in Base 3

Best Known (168−35, 168, s)-Nets in Base 3

(168−35, 168, 688)-Net over F₃ — Constructive and digital

Digital (133, 168, 688)-net over F₃, using

trace code for nets [i] based on digital (7, 42, 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]

(168−35, 168, 1679)-Net over F₃ — Digital

Digital (133, 168, 1679)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶⁸, 1679, F₃, 35) (dual of [1679, 1511, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁶⁸, 2214, F₃, 35) (dual of [2214, 2046, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(30) [i] based on
    1. 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]
    2. linear OA(3¹⁴¹, 2187, F₃, 31) (dual of [2187, 2046, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    3. linear OA(3⁶, 27, F₃, 3) (dual of [27, 21, 4]-code or 27-cap in PG(5,3)), using
      - discarding factors / shortening the dual code based on linear OA(3⁶, 48, F₃, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
        large caps in PG(5,Â b) [i]

(168−35, 168, 174537)-Net in Base 3 — Upper bound on s

There is no (133, 168, 174538)-net in base 3, because

1 times m-reduction [i] would yield (133, 167, 174538)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 47 782538 981920 303810 337096 850355 065985 308160 525886 992174 568426 990320 666091 959797 > 3¹⁶⁷ [i]