MinT - Best Known (159−34, 159, s)-Nets in Base 3

Best Known (159−34, 159, s)-Nets in Base 3

(159−34, 159, 640)-Net over F₃ — Constructive and digital

Digital (125, 159, 640)-net over F₃, using

1 times m-reduction [i] based on digital (125, 160, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 40, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(159−34, 159, 1422)-Net over F₃ — Digital

Digital (125, 159, 1422)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵⁹, 1422, F₃, 34) (dual of [1422, 1263, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁵⁹, 2205, F₃, 34) (dual of [2205, 2046, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(30) [i] based on
    1. linear OA(3¹⁵⁵, 2187, F₃, 34) (dual of [2187, 2032, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [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(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]

(159−34, 159, 104071)-Net in Base 3 — Upper bound on s

There is no (125, 159, 104072)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7282 627289 812898 821447 270090 897722 142576 996384 351110 054331 246424 768029 433361 > 3¹⁵⁹ [i]