MinT - Best Known (179−18, 179, s)-Nets in Base 3

Best Known (179−18, 179, s)-Nets in Base 3

(179−18, 179, 531449)-Net over F₃ — Constructive and digital

Digital (161, 179, 531449)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 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 (150, 168, 531441)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶⁸, 531441, F₃, 18, 18) (dual of [(531441, 18), 9565770, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(3¹⁶⁸, 4782969, F₃, 18) (dual of [4782969, 4782801, 19]-code), using
      - 1 times truncation [i] based on linear OA(3¹⁶⁹, 4782970, F₃, 19) (dual of [4782970, 4782801, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 3²⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(179−18, 179, 1594345)-Net over F₃ — Digital

Digital (161, 179, 1594345)-net over F₃, using

3¹ times duplication [i] based on digital (160, 178, 1594345)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷⁸, 1594345, F₃, 3, 18) (dual of [(1594345, 3), 4782857, 19]-NRT-code), using
  - OOA 3-folding [i] based on linear OA(3¹⁷⁸, 4783035, F₃, 18) (dual of [4783035, 4782857, 19]-code), using
    - constructionÂ X applied to C([0,9]) âŠ‚ C([0,6]) [i] based on
      1. linear OA(3¹⁶⁹, 4782970, F₃, 19) (dual of [4782970, 4782801, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 3²⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      2. linear OA(3¹¹³, 4782970, F₃, 13) (dual of [4782970, 4782857, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970Â | 3²⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
      3. linear OA(3⁹, 65, F₃, 4) (dual of [65, 56, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁹, 80, F₃, 4) (dual of [80, 71, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(179−18, 179, large)-Net in Base 3 — Upper bound on s

There is no (161, 179, large)-net in base 3, because

16 times m-reduction [i] would yield (161, 163, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]