MinT - Best Known (44, 59, s)-Nets in Base 3

Best Known (44, 59, s)-Nets in Base 3

(44, 59, 192)-Net over F₃ — Constructive and digital

Digital (44, 59, 192)-net over F₃, using

1 times m-reduction [i] based on digital (44, 60, 192)-net over F₃, using
- trace code for nets [i] based on digital (4, 20, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]

(44, 59, 370)-Net over F₃ — Digital

Digital (44, 59, 370)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵⁹, 370, F₃, 15) (dual of [370, 311, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁵⁹, 375, F₃, 15) (dual of [375, 316, 16]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(12) [i] based on
    1. linear OA(3⁵⁸, 365, F₃, 16) (dual of [365, 307, 17]-code), using an extension C_e(15) of the narrow-sense BCH-code C(I) with length 364Â | 3⁶âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. linear OA(3⁴⁹, 365, F₃, 13) (dual of [365, 316, 14]-code), using an extension C_e(12) of the narrow-sense BCH-code C(I) with length 364Â | 3⁶âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    3. linear OA(3¹, 10, F₃, 1) (dual of [10, 9, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(44, 59, 15170)-Net in Base 3 — Upper bound on s

There is no (44, 59, 15171)-net in base 3, because

1 times m-reduction [i] would yield (44, 58, 15171)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4711 735359 937828 834593 474819 > 3⁵⁸ [i]