MinT - Best Known (161, 161+29, s)-Nets in Base 3

Best Known (161, 161+29, s)-Nets in Base 3

(161, 161+29, 1480)-Net over F₃ — Constructive and digital

Digital (161, 190, 1480)-net over F₃, using

t-expansion [i] based on digital (160, 190, 1480)-net over F₃, using
- 2 times m-reduction [i] based on digital (160, 192, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 48, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(161, 161+29, 11920)-Net over F₃ — Digital

Digital (161, 190, 11920)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹⁰, 11920, F₃, 29) (dual of [11920, 11730, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁹⁰, 19703, F₃, 29) (dual of [19703, 19513, 30]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(3¹⁸, 20, F₃, 14) (dual of [20, 2, 15]-code), using
      - repeating each code word 5 times [i] based on linear OA(3², 4, F₃, 2) (dual of [4, 2, 3]-code or 4-arc in PG(1,3)), using
        extended Reedâ€“Solomon code RS_e(2,3) [i]
        Hamming code H(2,3) [i]
        Simplex code S(2,3) [i]
        the Tetracode [i]
    2. linear OA(3¹⁷², 19683, F₃, 29) (dual of [19683, 19511, 30]-code), using
      - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(161, 161+29, 8347645)-Net in Base 3 — Upper bound on s

There is no (161, 190, 8347646)-net in base 3, because

1 times m-reduction [i] would yield (161, 189, 8347646)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 499400 841791 836005 421869 670306 196494 090926 206926 497945 854203 734549 324975 072614 912360 333621 > 3¹⁸⁹ [i]