MinT - Best Known (175−24, 175, s)-Nets in Base 3

Best Known (175−24, 175, s)-Nets in Base 3

(175−24, 175, 4930)-Net over F₃ — Constructive and digital

Digital (151, 175, 4930)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 10)-net over F₃, using
  - net from sequence [i] based on digital (3, 9)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 3 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (136, 160, 4920)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁶⁰, 4920, F₃, 24, 24) (dual of [(4920, 24), 117920, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(3¹⁶⁰, 59040, F₃, 24) (dual of [59040, 58880, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁶⁰, 59049, F₃, 24) (dual of [59049, 58889, 25]-code), using
        1 times truncation [i] based on linear OA(3¹⁶¹, 59050, F₃, 25) (dual of [59050, 58889, 26]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 59050Â | 3²⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(175−24, 175, 29552)-Net over F₃ — Digital

Digital (151, 175, 29552)-net over F₃, using

3¹ times duplication [i] based on digital (150, 174, 29552)-net over F₃, using
- embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁷⁴, 29552, F₃, 2, 24) (dual of [(29552, 2), 58930, 25]-NRT-code), using
  - OOA 2-folding [i] based on linear OA(3¹⁷⁴, 59104, F₃, 24) (dual of [59104, 58930, 25]-code), using
    - constructionÂ X with VarÅ¡amov bound [i] based on
      1. linear OA(3¹⁷⁰, 59099, F₃, 24) (dual of [59099, 58929, 25]-code), using
        constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
        linear OA(3¹⁶¹, 59050, F₃, 25) (dual of [59050, 58889, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050Â | 3²⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
        linear OA(3¹²¹, 59050, F₃, 19) (dual of [59050, 58929, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050Â | 3²⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
        linear OA(3⁹, 49, F₃, 4) (dual of [49, 40, 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]
      2. linear OA(3¹⁷⁰, 59100, F₃, 20) (dual of [59100, 58930, 21]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 3¹⁷⁰Â > V_b^sâˆ’1(kâˆ’1)Â = 19 647663 364515 428307 323077 719525 878234 129146 170108 217591 012313 925788 230854 612155 [i]
      3. linear OA(3³, 4, F₃, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,3) or 4-cap in PG(2,3)), using
        dual of repetition code with length 4 [i]
        oval in PG(2,Â 3) [i]

(175−24, 175, large)-Net in Base 3 — Upper bound on s

There is no (151, 175, large)-net in base 3, because

22 times m-reduction [i] would yield (151, 153, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (175−24, 175, s)-Nets in Base 3

(175−24, 175, 4930)-Net over F3 — Constructive and digital

(175−24, 175, 29552)-Net over F3 — Digital

(175−24, 175, large)-Net in Base 3 — Upper bound on s

(175−24, 175, 4930)-Net over F₃ — Constructive and digital

(175−24, 175, 29552)-Net over F₃ — Digital