MinT - Best Known (231−14, 231, s)-Nets in Base 3

Best Known (231−14, 231, s)-Nets in Base 3

(231−14, 231, 4793538)-Net over F₃ — Constructive and digital

Digital (217, 231, 4793538)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (12, 19, 54)-net over F₃, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 5, 20)-net over F₃, using
      - net defined by OOA [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]
    2. digital (7, 14, 34)-net over F₃, using
      - linear code embedding found in a computer search [i]
2. digital (198, 212, 4793484)-net over F₃, using
  - trace code for nets [i] based on digital (39, 53, 1198371)-net over F₈₁, using
    - net defined by OOA [i] based on linear OOA(81⁵³, 1198371, F₈₁, 14, 14) (dual of [(1198371, 14), 16777141, 15]-NRT-code), using
      - OA 7-folding and stacking [i] based on linear OA(81⁵³, 8388597, F₈₁, 14) (dual of [8388597, 8388544, 15]-code), using
        discarding factors / shortening the dual code based on linear OA(81⁵³, large, F₈₁, 14) (dual of [large, large−53, 15]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 21523360Â | 81⁴âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(231−14, 231, large)-Net over F₃ — Digital

Digital (217, 231, large)-net over F₃, using

3³ times duplication [i] based on digital (214, 228, large)-net over F₃, using
- t-expansion [i] based on digital (209, 228, large)-net over F₃, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²²⁸, large, F₃, 19) (dual of [large, large−228, 20]-code), using
    - 47 times code embedding in larger space [i] based on linear OA(3¹⁸¹, large, F₃, 19) (dual of [large, large−181, 20]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(231−14, 231, large)-Net in Base 3 — Upper bound on s

There is no (217, 231, large)-net in base 3, because

12 times m-reduction [i] would yield (217, 219, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (231−14, 231, s)-Nets in Base 3

(231−14, 231, 4793538)-Net over F3 — Constructive and digital

(231−14, 231, large)-Net over F3 — Digital

(231−14, 231, large)-Net in Base 3 — Upper bound on s

(231−14, 231, 4793538)-Net over F₃ — Constructive and digital

(231−14, 231, large)-Net over F₃ — Digital