MinT - Best Known (137−12, 137, s)-Nets in Base 3

Best Known (137−12, 137, s)-Nets in Base 3

(137−12, 137, 1398153)-Net over F₃ — Constructive and digital

Digital (125, 137, 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, 120, 1398100)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹²⁰, 1398100, F₃, 12, 12) (dual of [(1398100, 12), 16777080, 13]-NRT-code), using
    - OA 6-folding and stacking [i] based on linear OA(3¹²⁰, 8388600, F₃, 12) (dual of [8388600, 8388480, 13]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹²⁰, large, F₃, 12) (dual of [large, large−120, 13]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(137−12, 137, 6979026)-Net over F₃ — Digital

Digital (125, 137, 6979026)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³⁷, 6979026, F₃, 12) (dual of [6979026, 6978889, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3¹³⁷, large, F₃, 12) (dual of [large, large−137, 13]-code), using
  - 17 times code embedding in larger space [i] based on linear OA(3¹²⁰, large, F₃, 12) (dual of [large, large−120, 13]-code), using
    - the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,12], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 13 [i]

(137−12, 137, large)-Net in Base 3 — Upper bound on s

There is no (125, 137, large)-net in base 3, because

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

Best Known (137−12, 137, s)-Nets in Base 3

(137−12, 137, 1398153)-Net over F3 — Constructive and digital

(137−12, 137, 6979026)-Net over F3 — Digital

(137−12, 137, large)-Net in Base 3 — Upper bound on s

(137−12, 137, 1398153)-Net over F₃ — Constructive and digital

(137−12, 137, 6979026)-Net over F₃ — Digital