MinT - Best Known (141, 141+14, s)-Nets in Base 3

Best Known (141, 141+14, s)-Nets in Base 3

(141, 141+14, 1198425)-Net over F₃ — Constructive and digital

Digital (141, 155, 1198425)-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 (122, 136, 1198371)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹³⁶, 1198371, F₃, 14, 14) (dual of [(1198371, 14), 16777058, 15]-NRT-code), using
    - OA 7-folding and stacking [i] based on linear OA(3¹³⁶, 8388597, F₃, 14) (dual of [8388597, 8388461, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹³⁶, large, F₃, 14) (dual of [large, large−136, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(141, 141+14, 4194365)-Net over F₃ — Digital

Digital (141, 155, 4194365)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁵⁵, 4194365, F₃, 2, 14) (dual of [(4194365, 2), 8388575, 15]-NRT-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OOA(3¹⁹, 64, F₃, 2, 7) (dual of [(64, 2), 109, 8]-NRT-code), using
    - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁹, 64, F₃, 7) (dual of [64, 45, 8]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹⁹, 85, F₃, 7) (dual of [85, 66, 8]-code), using
        constructionÂ X applied to C([0,3]) âŠ‚ C([1,3]) [i] based on
        linear OA(3¹⁷, 82, F₃, 7) (dual of [82, 65, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 82Â | 3⁸âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(3¹⁶, 82, F₃, 4) (dual of [82, 66, 5]-code), using the narrow-sense BCH-code C(I) with length 82Â | 3⁸âˆ’1, defining interval IÂ =Â [1,3], and minimum distance dÂ â‰¥ |{1,2,3}|Â +Â |{0,39,78,35}âˆ–{39,35}|Â =Â 5 (general Roos-bound) [i]
        linear OA(3², 3, F₃, 2) (dual of [3, 1, 3]-code or 3-arc in PG(1,3)), using
        dual of repetition code with length 3 [i]
        Reedâ€“Solomon code RS(1,3) [i]
  2. linear OOA(3¹³⁶, 4194301, F₃, 2, 14) (dual of [(4194301, 2), 8388466, 15]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(3¹³⁶, 8388602, F₃, 14) (dual of [8388602, 8388466, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(3¹³⁶, large, F₃, 14) (dual of [large, large−136, 15]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(141, 141+14, large)-Net in Base 3 — Upper bound on s

There is no (141, 155, large)-net in base 3, because

12 times m-reduction [i] would yield (141, 143, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (141, 141+14, s)-Nets in Base 3

(141, 141+14, 1198425)-Net over F3 — Constructive and digital

(141, 141+14, 4194365)-Net over F3 — Digital

(141, 141+14, large)-Net in Base 3 — Upper bound on s

(141, 141+14, 1198425)-Net over F₃ — Constructive and digital

(141, 141+14, 4194365)-Net over F₃ — Digital