MinT - Best Known (131, 131+13, s)-Nets in Base 3

Best Known (131, 131+13, s)-Nets in Base 3

(131, 131+13, 1398264)-Net over F₃ — Constructive and digital

Digital (131, 144, 1398264)-net over F₃, using

3¹ times duplication [i] based on digital (130, 143, 1398264)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (16, 22, 164)-net over F₃, using
    - trace code for nets [i] based on digital (5, 11, 82)-net over F₉, using
      - base reduction for projective spaces (embedding PG(5,81) in PG(10,9)) for nets [i] based on digital (0, 6, 82)-net over F₈₁, using
        net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
        Niederreiter sequence [i]
  2. digital (108, 121, 1398100)-net over F₃, using
    - net defined by OOA [i] based on linear OOA(3¹²¹, 1398100, F₃, 13, 13) (dual of [(1398100, 13), 18175179, 14]-NRT-code), using
      - OOA 6-folding and stacking with additional row [i] based on linear OA(3¹²¹, 8388601, F₃, 13) (dual of [8388601, 8388480, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(3¹²¹, large, F₃, 13) (dual of [large, large−121, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(131, 131+13, 4194568)-Net over F₃ — Digital

Digital (131, 144, 4194568)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁴⁴, 4194568, F₃, 2, 13) (dual of [(4194568, 2), 8388992, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3²³, 267, F₃, 2, 6) (dual of [(267, 2), 511, 7]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²³, 267, F₃, 6) (dual of [267, 244, 7]-code), using
      - 13 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 0, 0, 1, 9 times 0) [i] based on linear OA(3²¹, 252, F₃, 6) (dual of [252, 231, 7]-code), using
        constructionÂ XX applied to C₁Â =Â C([241,3]), C₂Â =Â C([0,4]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,3]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([241,4]) [i] based on
        linear OA(3¹⁶, 242, F₃, 5) (dual of [242, 226, 6]-code), using the primitive BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â {−1,0,1,2,3}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(3¹⁶, 242, F₃, 5) (dual of [242, 226, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
        linear OA(3²¹, 242, F₃, 6) (dual of [242, 221, 7]-code), using the primitive BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â {−1,0,…,4}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(3¹¹, 242, F₃, 4) (dual of [242, 231, 5]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242Â = 3⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(3⁰, 5, F₃, 0) (dual of [5, 5, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁰, s, F₃, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
        linear OA(3⁰, 5, F₃, 0) (dual of [5, 5, 1]-code) (see above)
  2. linear OOA(3¹²¹, 4194301, F₃, 2, 13) (dual of [(4194301, 2), 8388481, 14]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹²¹, 8388602, F₃, 13) (dual of [8388602, 8388481, 14]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹²¹, large, F₃, 13) (dual of [large, large−121, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(131, 131+13, large)-Net in Base 3 — Upper bound on s

There is no (131, 144, large)-net in base 3, because

11 times m-reduction [i] would yield (131, 133, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (131, 131+13, s)-Nets in Base 3

(131, 131+13, 1398264)-Net over F3 — Constructive and digital

(131, 131+13, 4194568)-Net over F3 — Digital

(131, 131+13, large)-Net in Base 3 — Upper bound on s

(131, 131+13, 1398264)-Net over F₃ — Constructive and digital

(131, 131+13, 4194568)-Net over F₃ — Digital