MinT - Best Known (52−16, 52, s)-Nets in Base 3

Best Known (52−16, 52, s)-Nets in Base 3

(52−16, 52, 114)-Net over F₃ — Constructive and digital

Digital (36, 52, 114)-net over F₃, using

3¹ times duplication [i] based on digital (35, 51, 114)-net over F₃, using
- trace code for nets [i] based on digital (1, 17, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(52−16, 52, 155)-Net over F₃ — Digital

Digital (36, 52, 155)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁵², 155, F₃, 16) (dual of [155, 103, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁵², 249, F₃, 16) (dual of [249, 197, 17]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(13) [i] based on
    1. linear OA(3⁵¹, 243, F₃, 16) (dual of [243, 192, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]
    2. linear OA(3⁴⁶, 243, F₃, 14) (dual of [243, 197, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
    3. linear OA(3¹, 6, F₃, 1) (dual of [6, 5, 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]

(52−16, 52, 2369)-Net in Base 3 — Upper bound on s

There is no (36, 52, 2370)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 481404 809202 376053 247569 > 3⁵² [i]