MinT - Best Known (167−35, 167, s)-Nets in Base 3

Best Known (167−35, 167, s)-Nets in Base 3

(167−35, 167, 640)-Net over F₃ — Constructive and digital

Digital (132, 167, 640)-net over F₃, using

t-expansion [i] based on digital (131, 167, 640)-net over F₃, using
- 1 times m-reduction [i] based on digital (131, 168, 640)-net over F₃, using
  - trace code for nets [i] based on digital (5, 42, 160)-net over F₈₁, using
    - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
        a function field by SÃ©mirat [i]

(167−35, 167, 1623)-Net over F₃ — Digital

Digital (132, 167, 1623)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁶⁷, 1623, F₃, 35) (dual of [1623, 1456, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁶⁷, 2207, F₃, 35) (dual of [2207, 2040, 36]-code), using
  - constructionÂ X applied to C_e(34) âŠ‚ C_e(30) [i] based on
    1. linear OA(3¹⁶², 2187, F₃, 35) (dual of [2187, 2025, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]
    2. linear OA(3¹⁴¹, 2187, F₃, 31) (dual of [2187, 2046, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [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]

(167−35, 167, 163613)-Net in Base 3 — Upper bound on s

There is no (132, 167, 163614)-net in base 3, because

1 times m-reduction [i] would yield (132, 166, 163614)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 15 926991 794014 142474 295600 391695 341950 355117 155456 032182 967762 355314 745244 674077 > 3¹⁶⁶ [i]