MinT - Best Known (115−20, 115, s)-Nets in Base 3

Best Known (115−20, 115, s)-Nets in Base 3

(115−20, 115, 688)-Net over F₃ — Constructive and digital

Digital (95, 115, 688)-net over F₃, using

t-expansion [i] based on digital (94, 115, 688)-net over F₃, using
- 1 times m-reduction [i] based on digital (94, 116, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 29, 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]

(115−20, 115, 3954)-Net over F₃ — Digital

Digital (95, 115, 3954)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁵, 3954, F₃, 20) (dual of [3954, 3839, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁵, 6572, F₃, 20) (dual of [6572, 6457, 21]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹⁰, 11, F₃, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,3)), using
      - dual of repetition code with length 11 [i]
    2. linear OA(3¹⁰⁵, 6561, F₃, 20) (dual of [6561, 6456, 21]-code), using
      - an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]

(115−20, 115, 694759)-Net in Base 3 — Upper bound on s

There is no (95, 115, 694760)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 395105 472006 356742 002288 037854 469660 848886 851548 076401 > 3¹¹⁵ [i]