MinT - Best Known (249−41, 249, s)-Nets in Base 3

Best Known (249−41, 249, s)-Nets in Base 3

(249−41, 249, 1487)-Net over F₃ — Constructive and digital

Digital (208, 249, 1487)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 21, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (187, 228, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 57, 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]

(249−41, 249, 9449)-Net over F₃ — Digital

Digital (208, 249, 9449)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁴⁹, 9449, F₃, 2, 41) (dual of [(9449, 2), 18649, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3²⁴⁹, 9853, F₃, 2, 41) (dual of [(9853, 2), 19457, 42]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²⁴⁹, 19706, F₃, 41) (dual of [19706, 19457, 42]-code), using
    - 1 times code embedding in larger space [i] based on linear OA(3²⁴⁸, 19705, F₃, 41) (dual of [19705, 19457, 42]-code), using
      - constructionÂ X applied to C_e(40) âŠ‚ C_e(37) [i] based on
        linear OA(3²⁴⁴, 19683, F₃, 41) (dual of [19683, 19439, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
        linear OA(3²²⁶, 19683, F₃, 38) (dual of [19683, 19457, 39]-code), using an extension C_e(37) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,37], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 38 [i]
        linear OA(3⁴, 22, F₃, 2) (dual of [22, 18, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(249−41, 249, 3424343)-Net in Base 3 — Upper bound on s

There is no (208, 249, 3424344)-net in base 3, because

1 times m-reduction [i] would yield (208, 248, 3424344)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21187 147116 066778 512309 461621 395578 671770 476823 281181 796527 305724 641160 902765 022890 210510 554049 708069 546669 617441 001153 > 3²⁴⁸ [i]