MinT - Best Known (118−23, 118, s)-Nets in Base 3

Best Known (118−23, 118, s)-Nets in Base 3

(118−23, 118, 640)-Net over F₃ — Constructive and digital

Digital (95, 118, 640)-net over F₃, using

2 times m-reduction [i] based on digital (95, 120, 640)-net over F₃, using
- trace code for nets [i] based on digital (5, 30, 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]

(118−23, 118, 1956)-Net over F₃ — Digital

Digital (95, 118, 1956)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹¹⁸, 1956, F₃, 23) (dual of [1956, 1838, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹¹⁸, 2208, F₃, 23) (dual of [2208, 2090, 24]-code), using
  - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
    1. linear OA(3¹¹³, 2188, F₃, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
    2. linear OA(3⁸⁵, 2188, F₃, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188Â | 3¹⁴âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (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]

(118−23, 118, 291610)-Net in Base 3 — Upper bound on s

There is no (95, 118, 291611)-net in base 3, because

1 times m-reduction [i] would yield (95, 117, 291611)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 66 556299 301714 190430 818988 564747 683474 099757 752240 729211 > 3¹¹⁷ [i]