MinT - Best Known (138−13, 138, s)-Nets in Base 3

Best Known (138−13, 138, s)-Nets in Base 3

(138−13, 138, 1398153)-Net over F₃ — Constructive and digital

Digital (125, 138, 1398153)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 17, 53)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁷, 53, F₃, 6, 6) (dual of [(53, 6), 301, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(3¹⁷, 53, F₃, 5, 6) (dual of [(53, 5), 248, 7]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(3⁵, 20, F₃, 5, 3) (dual of [(20, 5), 95, 4]-NRT-code), using
        appending 2 arbitrary columns [i] based on linear OOA(3⁵, 20, F₃, 3, 3) (dual of [(20, 3), 55, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(3⁵, 20, F₃, 2, 3) (dual of [(20, 2), 35, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(3⁵, 20, F₃, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
        doubling the cap [i] based on linear OA(3⁴, 10, F₃, 3) (dual of [10, 6, 4]-code or 10-cap in PG(3,3)), using
        ovoid in PG(3,Â 3) [i]
        linear OOA(3¹², 33, F₃, 5, 6) (dual of [(33, 5), 153, 7]-NRT-code), using
        extracting embedded OOA [i] based on digital (6, 12, 33)-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]

(138−13, 138, 4194388)-Net over F₃ — Digital

Digital (125, 138, 4194388)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹³⁸, 4194388, F₃, 2, 13) (dual of [(4194388, 2), 8388638, 14]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁷, 87, F₃, 2, 6) (dual of [(87, 2), 157, 7]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁷, 87, F₃, 6) (dual of [87, 70, 7]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁷, 88, F₃, 6) (dual of [88, 71, 7]-code), using
        constructionÂ XX applied to C₁Â =Â C({0,1,2,53}), C₂Â =Â C([0,4]), C₃Â =Â C₁Â +Â C₂Â =Â C([0,2]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C({0,1,2,4,53}) [i] based on
        linear OA(3¹³, 80, F₃, 5) (dual of [80, 67, 6]-code), using the primitive cyclic code C(A) with length 80Â = 3⁴âˆ’1, defining set AÂ =Â {0,1,2,53}, and minimum distance dÂ â‰¥ |{−1,0,1,2,3}|+1Â =Â 6 (BCH-bound) [i]
        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]
        linear OA(3¹⁷, 80, F₃, 6) (dual of [80, 63, 7]-code), using the primitive cyclic code C(A) with length 80Â = 3⁴âˆ’1, defining set AÂ =Â {0,1,2,4,53}, and minimum distance dÂ â‰¥ |{−1,0,…,4}|+1Â =Â 7 (BCH-bound) [i]
        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]
        linear OA(3⁰, 4, F₃, 0) (dual of [4, 4, 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, 4, F₃, 0) (dual of [4, 4, 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]

(138−13, 138, large)-Net in Base 3 — Upper bound on s

There is no (125, 138, large)-net in base 3, because

11 times m-reduction [i] would yield (125, 127, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (138−13, 138, s)-Nets in Base 3

(138−13, 138, 1398153)-Net over F3 — Constructive and digital

(138−13, 138, 4194388)-Net over F3 — Digital

(138−13, 138, large)-Net in Base 3 — Upper bound on s

(138−13, 138, 1398153)-Net over F₃ — Constructive and digital

(138−13, 138, 4194388)-Net over F₃ — Digital