MinT - Best Known (38, 50, s)-Nets in Base 3

Best Known (38, 50, s)-Nets in Base 3

(38, 50, 328)-Net over F₃ — Constructive and digital

Digital (38, 50, 328)-net over F₃, using

3² times duplication [i] based on digital (36, 48, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 12, 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]

(38, 50, 484)-Net over F₃ — Digital

Digital (38, 50, 484)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵⁰, 484, F₃, 12) (dual of [484, 434, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁵⁰, 742, F₃, 12) (dual of [742, 692, 13]-code), using
  - constructionÂ X applied to C_e(12) âŠ‚ C_e(9) [i] based on
    1. 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]
    2. linear OA(3³⁷, 729, F₃, 10) (dual of [729, 692, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
    3. linear OA(3¹, 13, F₃, 1) (dual of [13, 12, 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]

(38, 50, 14159)-Net in Base 3 — Upper bound on s

There is no (38, 50, 14160)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 718189 023276 473915 055585 > 3⁵⁰ [i]