MinT - Best Known (109, 109+25, s)-Nets in Base 3

Best Known (109, 109+25, s)-Nets in Base 3

(109, 109+25, 688)-Net over F₃ — Constructive and digital

Digital (109, 134, 688)-net over F₃, using

2 times m-reduction [i] based on digital (109, 136, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 34, 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]

(109, 109+25, 3279)-Net over F₃ — Digital

Digital (109, 134, 3279)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹³⁴, 3279, F₃, 2, 25) (dual of [(3279, 2), 6424, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹³⁴, 3291, F₃, 2, 25) (dual of [(3291, 2), 6448, 26]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹³⁴, 6582, F₃, 25) (dual of [6582, 6448, 26]-code), using
    - constructionÂ X applied to C([0,12]) âŠ‚ C([0,10]) [i] based on
      1. linear OA(3¹²⁹, 6562, F₃, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      2. linear OA(3¹¹³, 6562, F₃, 21) (dual of [6562, 6449, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      3. linear OA(3⁵, 20, F₃, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
        doubling the cap [i] based on linear OA(3⁴, 10, F₃, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), using
        ovoid in PG(3,Â 3) [i]

(109, 109+25, 513352)-Net in Base 3 — Upper bound on s

There is no (109, 134, 513353)-net in base 3, because

1 times m-reduction [i] would yield (109, 133, 513353)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2865 055116 578613 442231 940965 311378 617767 800185 354663 243492 710833 > 3¹³³ [i]