MinT - Best Known (94, 116, s)-Nets in Base 3

Best Known (94, 116, s)-Nets in Base 3

(94, 116, 688)-Net over F₃ — Constructive and digital

Digital (94, 116, 688)-net over F₃, using

trace code for nets [i] based on digital (7, 29, 172)-net over F₈₁, using
- net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
    - a function field by SÃ©mirat [i]

(94, 116, 2872)-Net over F₃ — Digital

Digital (94, 116, 2872)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹¹⁶, 2872, F₃, 2, 22) (dual of [(2872, 2), 5628, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹¹⁶, 3287, F₃, 2, 22) (dual of [(3287, 2), 6458, 23]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹¹⁶, 6574, F₃, 22) (dual of [6574, 6458, 23]-code), using
    - constructionÂ X applied to C_e(21) âŠ‚ C_e(18) [i] based on
      1. linear OA(3¹¹³, 6561, F₃, 22) (dual of [6561, 6448, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      2. linear OA(3⁹⁷, 6561, F₃, 19) (dual of [6561, 6464, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
      3. linear OA(3³, 13, F₃, 2) (dual of [13, 10, 3]-code), using
        Hamming code H(3,3) [i]

(94, 116, 263892)-Net in Base 3 — Upper bound on s

There is no (94, 116, 263893)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22 185584 837668 934301 250110 510350 098361 906932 475077 786059 > 3¹¹⁶ [i]