MinT - Best Known (135, 135+31, s)-Nets in Base 3

Best Known (135, 135+31, s)-Nets in Base 3

(135, 135+31, 688)-Net over F₃ — Constructive and digital

Digital (135, 166, 688)-net over F₃, using

t-expansion [i] based on digital (133, 166, 688)-net over F₃, using
- 2 times m-reduction [i] based on digital (133, 168, 688)-net over F₃, using
  - trace code for nets [i] based on digital (7, 42, 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]

(135, 135+31, 3291)-Net over F₃ — Digital

Digital (135, 166, 3291)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁶⁶, 3291, F₃, 2, 31) (dual of [(3291, 2), 6416, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3¹⁶⁶, 6582, F₃, 31) (dual of [6582, 6416, 32]-code), using
  - constructionÂ X applied to C([0,15]) âŠ‚ C([0,13]) [i] based on
    1. linear OA(3¹⁶¹, 6562, F₃, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,15], and minimum distance dÂ â‰¥ |{−15,−14,…,15}|+1Â =Â 32 (BCH-bound) [i]
    2. linear OA(3¹⁴⁵, 6562, F₃, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562Â | 3¹⁶âˆ’1, defining interval IÂ =Â [0,13], and minimum distance dÂ â‰¥ |{−13,−12,…,13}|+1Â =Â 28 (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]

(135, 135+31, 568930)-Net in Base 3 — Upper bound on s

There is no (135, 166, 568931)-net in base 3, because

1 times m-reduction [i] would yield (135, 165, 568931)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 308970 641568 126407 810069 598912 110838 722508 667403 513052 087331 534645 837568 795251 > 3¹⁶⁵ [i]