MinT - Best Known (248−40, 248, s)-Nets in Base 3

Best Known (248−40, 248, s)-Nets in Base 3

(248−40, 248, 1492)-Net over F₃ — Constructive and digital

Digital (208, 248, 1492)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 24, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (184, 224, 1480)-net over F₃, using
  - trace code for nets [i] based on digital (16, 56, 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]

(248−40, 248, 9866)-Net over F₃ — Digital

Digital (208, 248, 9866)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²⁴⁸, 9866, F₃, 2, 40) (dual of [(9866, 2), 19484, 41]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3²⁴⁶, 9865, F₃, 2, 40) (dual of [(9865, 2), 19484, 41]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3²⁴⁶, 19730, F₃, 40) (dual of [19730, 19484, 41]-code), using
    - constructionÂ X applied to C_e(39) âŠ‚ C_e(33) [i] based on
      1. linear OA(3²³⁵, 19683, F₃, 40) (dual of [19683, 19448, 41]-code), using an extension C_e(39) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,39], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 40 [i]
      2. linear OA(3¹⁹⁹, 19683, F₃, 34) (dual of [19683, 19484, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 19682Â = 3⁹âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
      3. linear OA(3¹¹, 47, F₃, 5) (dual of [47, 36, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹¹, 85, F₃, 5) (dual of [85, 74, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(248−40, 248, 3424343)-Net in Base 3 — Upper bound on s

There is no (208, 248, 3424344)-net in base 3, because

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]