MinT - Best Known (89, 89+21, s)-Nets in Base 3

Best Known (89, 89+21, s)-Nets in Base 3

(89, 89+21, 640)-Net over F₃ — Constructive and digital

Digital (89, 110, 640)-net over F₃, using

2 times m-reduction [i] based on digital (89, 112, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 28, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(89, 89+21, 2147)-Net over F₃ — Digital

Digital (89, 110, 2147)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁰, 2147, F₃, 21) (dual of [2147, 2037, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁰, 2227, F₃, 21) (dual of [2227, 2117, 22]-code), using
  - constructionÂ X applied to C([0,10]) âŠ‚ C([0,7]) [i] based on
    1. linear OA(3⁹⁹, 2188, F₃, 21) (dual of [2188, 2089, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
    2. linear OA(3⁷¹, 2188, F₃, 15) (dual of [2188, 2117, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,7], and minimum distance dÂ â‰¥ |{−7,−6,…,7}|+1Â =Â 16 (BCH-bound) [i]
    3. linear OA(3¹¹, 39, F₃, 5) (dual of [39, 28, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹¹, 40, F₃, 5) (dual of [40, 29, 6]-code), using
        (u,Â uâˆ’v,Â u+v+w)-construction [i] based on
        linear OA(3¹, 13, F₃, 1) (dual of [13, 12, 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, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]
        linear OA(3⁷, 14, F₃, 5) (dual of [14, 7, 6]-code), using
        extended quadratic residue code Q_e(14,3) [i]

(89, 89+21, 359382)-Net in Base 3 — Upper bound on s

There is no (89, 110, 359383)-net in base 3, because

1 times m-reduction [i] would yield (89, 109, 359383)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10144 415295 783673 363510 379415 854846 329455 671735 802141 > 3¹⁰⁹ [i]

Best Known (89, 89+21, s)-Nets in Base 3

(89, 89+21, 640)-Net over F3 — Constructive and digital

(89, 89+21, 2147)-Net over F3 — Digital

(89, 89+21, 359382)-Net in Base 3 — Upper bound on s

(89, 89+21, 640)-Net over F₃ — Constructive and digital

(89, 89+21, 2147)-Net over F₃ — Digital