MinT - Best Known (87−18, 87, s)-Nets in Base 3

Best Known (87−18, 87, s)-Nets in Base 3

(87−18, 87, 464)-Net over F₃ — Constructive and digital

Digital (69, 87, 464)-net over F₃, using

t-expansion [i] based on digital (68, 87, 464)-net over F₃, using
- 1 times m-reduction [i] based on digital (68, 88, 464)-net over F₃, using
  - trace code for nets [i] based on digital (2, 22, 116)-net over F₈₁, using
    - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]

(87−18, 87, 1233)-Net over F₃ — Digital

Digital (69, 87, 1233)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁸⁷, 1233, F₃, 18) (dual of [1233, 1146, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁸⁷, 2192, F₃, 18) (dual of [2192, 2105, 19]-code), using
  - constructionÂ X applied to C([0,9]) âŠ‚ C([0,7]) [i] based on
    1. linear OA(3⁸⁵, 2188, F₃, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (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(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]

(87−18, 87, 84888)-Net in Base 3 — Upper bound on s

There is no (69, 87, 84889)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 323266 742129 180802 732171 643969 916551 955091 > 3⁸⁷ [i]