MinT - Best Known (140−34, 140, s)-Nets in Base 3

Best Known (140−34, 140, s)-Nets in Base 3

(140−34, 140, 400)-Net over F₃ — Constructive and digital

Digital (106, 140, 400)-net over F₃, using

trace code for nets [i] based on digital (1, 35, 100)-net over F₈₁, using
- net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
    - a function field by SÃ©mirat [i]

(140−34, 140, 728)-Net over F₃ — Digital

Digital (106, 140, 728)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁴⁰, 728, F₃, 34) (dual of [728, 588, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁴⁰, 760, F₃, 34) (dual of [760, 620, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(27) [i] based on
    1. linear OA(3¹³⁰, 729, F₃, 34) (dual of [729, 599, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(3¹⁰⁹, 729, F₃, 28) (dual of [729, 620, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    3. linear OA(3¹⁰, 31, F₃, 5) (dual of [31, 21, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁰, 36, F₃, 5) (dual of [36, 26, 6]-code), using
        (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
        linear OA(3¹, 12, F₃, 1) (dual of [12, 11, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OA(3³, 12, F₃, 2) (dual of [12, 9, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]
        linear OA(3⁶, 12, F₃, 5) (dual of [12, 6, 6]-code), using
        extended Golay code G_e(3) [i]

(140−34, 140, 30472)-Net in Base 3 — Upper bound on s

There is no (106, 140, 30473)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 266075 835322 804712 683098 495247 120965 146171 363351 658631 646817 499347 > 3¹⁴⁰ [i]

Best Known (140−34, 140, s)-Nets in Base 3

(140−34, 140, 400)-Net over F3 — Constructive and digital

(140−34, 140, 728)-Net over F3 — Digital

(140−34, 140, 30472)-Net in Base 3 — Upper bound on s

(140−34, 140, 400)-Net over F₃ — Constructive and digital

(140−34, 140, 728)-Net over F₃ — Digital