MinT - Best Known (52, 68, s)-Nets in Base 3

Best Known (52, 68, s)-Nets in Base 3

(52, 68, 400)-Net over F₃ — Constructive and digital

Digital (52, 68, 400)-net over F₃, using

trace code for nets [i] based on digital (1, 17, 100)-net over F₈₁, using
- net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
    - a function field by SÃ©mirat [i]

(52, 68, 568)-Net over F₃ — Digital

Digital (52, 68, 568)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁶⁸, 568, F₃, 16) (dual of [568, 500, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁶⁸, 754, F₃, 16) (dual of [754, 686, 17]-code), using
  - constructionÂ X applied to C_e(15) âŠ‚ C_e(10) [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(3⁴³, 729, F₃, 11) (dual of [729, 686, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 728Â = 3⁶âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    3. linear OA(3⁷, 25, F₃, 4) (dual of [25, 18, 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]

(52, 68, 21381)-Net in Base 3 — Upper bound on s

There is no (52, 68, 21382)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 278 178988 497875 143259 469531 502545 > 3⁶⁸ [i]