MinT - Best Known (103−22, 103, s)-Nets in Base 3

Best Known (103−22, 103, s)-Nets in Base 3

(103−22, 103, 464)-Net over F₃ — Constructive and digital

Digital (81, 103, 464)-net over F₃, using

t-expansion [i] based on digital (80, 103, 464)-net over F₃, using
- 1 times m-reduction [i] based on digital (80, 104, 464)-net over F₃, using
  - trace code for nets [i] based on digital (2, 26, 116)-net over F₈₁, using
    - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]

(103−22, 103, 1108)-Net over F₃ — Digital

Digital (81, 103, 1108)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁰³, 1108, F₃, 22) (dual of [1108, 1005, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹⁰³, 2205, F₃, 22) (dual of [2205, 2102, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(18) [i] based on
    1. linear OA(3⁹⁹, 2187, F₃, 22) (dual of [2187, 2088, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(3⁸⁵, 2187, F₃, 19) (dual of [2187, 2102, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 2186Â = 3⁷âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. linear OA(3⁴, 18, F₃, 2) (dual of [18, 14, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(3⁴, 40, F₃, 2) (dual of [40, 36, 3]-code), using
        Hamming code H(4,3) [i]

(103−22, 103, 72029)-Net in Base 3 — Upper bound on s

There is no (81, 103, 72030)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13 916117 725109 358615 093665 863268 739647 384669 679409 > 3¹⁰³ [i]