MinT - Best Known (115−24, 115, s)-Nets in Base 3

Best Known (115−24, 115, s)-Nets in Base 3

(115−24, 115, 600)-Net over F₃ — Constructive and digital

Digital (91, 115, 600)-net over F₃, using

1 times m-reduction [i] based on digital (91, 116, 600)-net over F₃, using
- trace code for nets [i] based on digital (4, 29, 150)-net over F₈₁, using
  - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
      - a function field by SÃ©mirat [i]

(115−24, 115, 1323)-Net over F₃ — Digital

Digital (91, 115, 1323)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁵, 1323, F₃, 24) (dual of [1323, 1208, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁵, 2192, F₃, 24) (dual of [2192, 2077, 25]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,10]) [i] based on
    1. linear OA(3¹¹³, 2188, F₃, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    2. 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]
    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]

(115−24, 115, 98785)-Net in Base 3 — Upper bound on s

There is no (91, 115, 98786)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 395617 081594 677552 682380 623223 098606 651363 960599 489913 > 3¹¹⁵ [i]