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

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

(131−13, 131, 1398119)-Net over F₃ — Constructive and digital

Digital (118, 131, 1398119)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 10, 19)-net over F₃, using
  - linear code embedding found in a computer search [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]

(131−13, 131, 3234516)-Net over F₃ — Digital

Digital (118, 131, 3234516)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹³¹, 3234516, F₃, 2, 13) (dual of [(3234516, 2), 6468901, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3¹³¹, 4194320, F₃, 2, 13) (dual of [(4194320, 2), 8388509, 14]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(3¹⁰, 19, F₃, 2, 6) (dual of [(19, 2), 28, 7]-NRT-code), using
      - extracting embedded OOA [i] based on digital (4, 10, 19)-net over F₃, using
        linear code embedding found in a computer search [i]
    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]

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

There is no (118, 131, large)-net in base 3, because

11 times m-reduction [i] would yield (118, 120, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]