MinT - Best Known (178, 178+43, s)-Nets in Base 3

Best Known (178, 178+43, s)-Nets in Base 3

(178, 178+43, 692)-Net over F₃ — Constructive and digital

Digital (178, 221, 692)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (0, 21, 4)-net over F₃, using
  - net from sequence [i] based on digital (0, 3)-sequence over F₃, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 0 and N(F) ≥ 4, using
      - the rational function field F₃(x) [i]
    - Niederreiter sequence [i]
2. digital (157, 200, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 50, 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]

(178, 178+43, 2893)-Net over F₃ — Digital

Digital (178, 221, 2893)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²¹, 2893, F₃, 43) (dual of [2893, 2672, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(3²²¹, 3281, F₃, 43) (dual of [3281, 3060, 44]-code), using
  - an extension C_e(42) of the narrow-sense BCH-code C(I) with length 3280Â | 3⁸âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]

(178, 178+43, 432369)-Net in Base 3 — Upper bound on s

There is no (178, 221, 432370)-net in base 3, because

1 times m-reduction [i] would yield (178, 220, 432370)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 926 157831 409540 382616 665976 373415 500263 107092 184480 307125 492764 041294 256423 364814 971355 581562 670076 695773 > 3²²⁰ [i]