MinT - Best Known (106, 106+24, s)-Nets in Base 3

Best Known (106, 106+24, s)-Nets in Base 3

(106, 106+24, 688)-Net over F₃ — Constructive and digital

Digital (106, 130, 688)-net over F₃, using

2 times m-reduction [i] based on digital (106, 132, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 33, 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]

(106, 106+24, 3289)-Net over F₃ — Digital

Digital (106, 130, 3289)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹³⁰, 3289, F₃, 2, 24) (dual of [(3289, 2), 6448, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹³⁰, 6578, F₃, 24) (dual of [6578, 6448, 25]-code), using
  - constructionÂ X applied to C_e(24) âŠ‚ C_e(21) [i] based on
    1. linear OA(3¹²⁹, 6561, F₃, 25) (dual of [6561, 6432, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 3⁸âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
    2. 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]
    3. linear OA(3¹, 17, F₃, 1) (dual of [17, 16, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹, s, F₃, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(106, 106+24, 390060)-Net in Base 3 — Upper bound on s

There is no (106, 130, 390061)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 106 112607 614702 737203 461828 591742 196038 010090 926710 370386 153073 > 3¹³⁰ [i]