MinT - Best Known (48−13, 48, s)-Nets in Base 3

Best Known (48−13, 48, s)-Nets in Base 3

(48−13, 48, 156)-Net over F₃ — Constructive and digital

Digital (35, 48, 156)-net over F₃, using

trace code for nets [i] based on digital (3, 16, 52)-net over F₂₇, using
- net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, using
    - a function field by SÃ©mirat [i]

(48−13, 48, 259)-Net over F₃ — Digital

Digital (35, 48, 259)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁴⁸, 259, F₃, 13) (dual of [259, 211, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁴⁸, 265, F₃, 13) (dual of [265, 217, 14]-code), using
  - constructionÂ X applied to C_e(12) âŠ‚ C_e(7) [i] based on
    1. linear OA(3⁴¹, 243, F₃, 13) (dual of [243, 202, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    2. linear OA(3²⁶, 243, F₃, 8) (dual of [243, 217, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. linear OA(3⁷, 22, F₃, 4) (dual of [22, 15, 5]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁷, 26, F₃, 4) (dual of [26, 19, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 26Â = 3³âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(48−13, 48, 8172)-Net in Base 3 — Upper bound on s

There is no (35, 48, 8173)-net in base 3, because

1 times m-reduction [i] would yield (35, 47, 8173)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 26600 446757 118314 916177 > 3⁴⁷ [i]