MinT - Best Known (148−29, 148, s)-Nets in Base 3

Best Known (148−29, 148, s)-Nets in Base 3

(148−29, 148, 688)-Net over F₃ — Constructive and digital

Digital (119, 148, 688)-net over F₃, using

t-expansion [i] based on digital (118, 148, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 37, 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]

(148−29, 148, 2138)-Net over F₃ — Digital

Digital (119, 148, 2138)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴⁸, 2138, F₃, 29) (dual of [2138, 1990, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁴⁸, 2202, F₃, 29) (dual of [2202, 2054, 30]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹⁴, 15, F₃, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,3)), using
      - dual of repetition code with length 15 [i]
    2. linear OA(3¹³⁴, 2187, F₃, 29) (dual of [2187, 2053, 30]-code), using
      - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(148−29, 148, 309159)-Net in Base 3 — Upper bound on s

There is no (119, 148, 309160)-net in base 3, because

1 times m-reduction [i] would yield (119, 147, 309160)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13703 870826 501736 269491 684012 183277 504603 819542 419635 292442 690441 172081 > 3¹⁴⁷ [i]