MinT - Best Known (61, 61+17, s)-Nets in Base 3

Best Known (61, 61+17, s)-Nets in Base 3

(61, 61+17, 464)-Net over F₃ — Constructive and digital

Digital (61, 78, 464)-net over F₃, using

3² times duplication [i] based on digital (59, 76, 464)-net over F₃, using
- trace code for nets [i] based on digital (2, 19, 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]

(61, 61+17, 1097)-Net over F₃ — Digital

Digital (61, 78, 1097)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3⁷⁸, 1097, F₃, 2, 17) (dual of [(1097, 2), 2116, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3⁷⁸, 2194, F₃, 17) (dual of [2194, 2116, 18]-code), using
  - constructionÂ X applied to C_e(16) âŠ‚ C_e(15) [i] based on
    1. linear OA(3⁷⁸, 2187, F₃, 17) (dual of [2187, 2109, 18]-code), using an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]
    2. linear OA(3⁷¹, 2187, F₃, 16) (dual of [2187, 2116, 17]-code), using an extension C_e(15) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [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]

(61, 61+17, 73605)-Net in Base 3 — Upper bound on s

There is no (61, 78, 73606)-net in base 3, because

1 times m-reduction [i] would yield (61, 77, 73606)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 474937 464082 505854 293905 626027 825105 > 3⁷⁷ [i]