MinT - Best Known (51, 51+13, s)-Nets in Base 3

Best Known (51, 51+13, s)-Nets in Base 3

(51, 51+13, 464)-Net over F₃ — Constructive and digital

Digital (51, 64, 464)-net over F₃, using

t-expansion [i] based on digital (50, 64, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 16, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(51, 51+13, 1316)-Net over F₃ — Digital

Digital (51, 64, 1316)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3⁶⁴, 1316, F₃, 13) (dual of [1316, 1252, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3⁶⁴, 2214, F₃, 13) (dual of [2214, 2150, 14]-code), using
  - constructionÂ X applied to C_e(12) âŠ‚ C_e(7) [i] based on
    1. linear OA(3⁵⁷, 2187, F₃, 13) (dual of [2187, 2130, 14]-code), using an extension C_e(12) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]
    2. linear OA(3³⁶, 2187, F₃, 8) (dual of [2187, 2151, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
    3. linear OA(3⁷, 27, F₃, 4) (dual of [27, 20, 5]-code), using
      - an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 26Â = 3³âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]

(51, 51+13, 153090)-Net in Base 3 — Upper bound on s

There is no (51, 64, 153091)-net in base 3, because

1 times m-reduction [i] would yield (51, 63, 153091)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 144561 279223 255645 154300 569757 > 3⁶³ [i]