MinT - Best Known (222−24, 222, s)-Nets in Base 3

Best Known (222−24, 222, s)-Nets in Base 3

(222−24, 222, 132868)-Net over F₃ — Constructive and digital

Digital (198, 222, 132868)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 14, 8)-net over F₃, using
  - net from sequence [i] based on digital (2, 7)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 2 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
2. digital (184, 208, 132860)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3²⁰⁸, 132860, F₃, 24, 24) (dual of [(132860, 24), 3188432, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(3²⁰⁸, 1594320, F₃, 24) (dual of [1594320, 1594112, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(3²⁰⁸, 1594323, F₃, 24) (dual of [1594323, 1594115, 25]-code), using
        1 times truncation [i] based on linear OA(3²⁰⁹, 1594324, F₃, 25) (dual of [1594324, 1594115, 26]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 1594324Â | 3²⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(222−24, 222, 531463)-Net over F₃ — Digital

Digital (198, 222, 531463)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3²²², 531463, F₃, 3, 24) (dual of [(531463, 3), 1594167, 25]-NRT-code), using
- strength reduction [i] based on linear OOA(3²²², 531463, F₃, 3, 25) (dual of [(531463, 3), 1594167, 26]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(3²²², 1594389, F₃, 25) (dual of [1594389, 1594167, 26]-code), using
    - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
      1. linear OA(3²⁰⁹, 1594324, F₃, 25) (dual of [1594324, 1594115, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324Â | 3²⁶âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      2. linear OA(3¹⁵⁷, 1594324, F₃, 19) (dual of [1594324, 1594167, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324Â | 3²⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      3. linear OA(3¹³, 65, F₃, 5) (dual of [65, 52, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹³, 80, F₃, 5) (dual of [80, 67, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(222−24, 222, large)-Net in Base 3 — Upper bound on s

There is no (198, 222, large)-net in base 3, because

22 times m-reduction [i] would yield (198, 200, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]