MinT - Best Known (49, 65, s)-Nets in Base 3

Best Known (49, 65, s)-Nets in Base 3

(49, 65, 328)-Net over F₃ — Constructive and digital

Digital (49, 65, 328)-net over F₃, using

3¹ times duplication [i] based on digital (48, 64, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 16, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]

(49, 65, 447)-Net over F₃ — Digital

Digital (49, 65, 447)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁶⁵, 447, F₃, 16) (dual of [447, 382, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁶⁵, 745, F₃, 16) (dual of [745, 680, 17]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(12) [i] based on
    1. linear OA(3⁶¹, 729, F₃, 16) (dual of [729, 668, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. linear OA(3⁴⁹, 729, F₃, 13) (dual of [729, 680, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(3⁴, 16, F₃, 2) (dual of [16, 12, 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]

(49, 65, 14159)-Net in Base 3 — Upper bound on s

There is no (49, 65, 14160)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 305398 816755 514233 361608 392705 > 3⁶⁵ [i]