MinT - Best Known (220−37, 220, s)-Nets in Base 3

Best Known (220−37, 220, s)-Nets in Base 3

(220−37, 220, 1480)-Net over F₃ — Constructive and digital

Digital (183, 220, 1480)-net over F₃, using

t-expansion [i] based on digital (181, 220, 1480)-net over F₃, using
- trace code for nets [i] based on digital (16, 55, 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]

(220−37, 220, 7723)-Net over F₃ — Digital

Digital (183, 220, 7723)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²²⁰, 7723, F₃, 2, 37) (dual of [(7723, 2), 15226, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²²⁰, 9848, F₃, 2, 37) (dual of [(9848, 2), 19476, 38]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²²⁰, 19696, F₃, 37) (dual of [19696, 19476, 38]-code), using
    - constructionÂ X applied to C_e(36) âŠ‚ C_e(33) [i] based on
      1. linear OA(3²¹⁷, 19683, F₃, 37) (dual of [19683, 19466, 38]-code), using an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]
      2. linear OA(3¹⁹⁹, 19683, F₃, 34) (dual of [19683, 19484, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]

(220−37, 220, 2410305)-Net in Base 3 — Upper bound on s

There is no (183, 220, 2410306)-net in base 3, because

1 times m-reduction [i] would yield (183, 219, 2410306)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 308 713009 995193 941628 153544 953566 239438 267317 381684 767014 109618 014097 966025 567254 694496 945952 018023 478837 > 3²¹⁹ [i]