MinT - Best Known (51, 65, s)-Nets in Base 3

Best Known (51, 65, s)-Nets in Base 3

(51, 65, 464)-Net over F₃ — Constructive and digital

Digital (51, 65, 464)-net over F₃, using

3¹ times duplication [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, 65, 1097)-Net over F₃ — Digital

Digital (51, 65, 1097)-net over F₃, using

3¹ times duplication [i] based on digital (50, 64, 1097)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁶⁴, 1097, F₃, 2, 14) (dual of [(1097, 2), 2130, 15]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3⁶⁴, 2194, F₃, 14) (dual of [2194, 2130, 15]-code), using
    - constructionÂ X applied to C_e(13) âŠ‚ C_e(12) [i] based on
      1. linear OA(3⁶⁴, 2187, F₃, 14) (dual of [2187, 2123, 15]-code), using an extension C_e(13) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 14 [i]
      2. 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]
      3. linear OA(3⁰, 7, F₃, 0) (dual of [7, 7, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(51, 65, 45523)-Net in Base 3 — Upper bound on s

There is no (51, 65, 45524)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 301402 215240 544614 387451 969457 > 3⁶⁵ [i]