MinT - Best Known (248−13, 248, s)-Nets in Base 3

Best Known (248−13, 248, s)-Nets in Base 3

(248−13, 248, 6990521)-Net over F₃ — Constructive and digital

Digital (235, 248, 6990521)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (59, 65, 2796221)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3⁶⁵, 2796221, F₃, 6, 6) (dual of [(2796221, 6), 16777261, 7]-NRT-code), using
    - appending kth column [i] based on linear OOA(3⁶⁵, 2796221, F₃, 5, 6) (dual of [(2796221, 5), 13981040, 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(3⁶⁰, 2796201, F₃, 5, 6) (dual of [(2796201, 5), 13980945, 7]-NRT-code), using
        OA 3-folding and stacking [i] based on linear OA(3⁶⁰, large, F₃, 6) (dual of [large, large−60, 7]-code), using
        the primitive narrow-sense BCH-code C(I) with length 14348906Â = 3¹⁵âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
2. digital (170, 183, 4194300)-net over F₃, using
  - trace code for nets [i] based on digital (48, 61, 1398100)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27⁶¹, 1398100, F₂₇, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
      - OOA 6-folding and stacking with additional row [i] based on linear OA(27⁶¹, 8388601, F₂₇, 13) (dual of [8388601, 8388540, 14]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁶¹, large, F₂₇, 13) (dual of [large, large−61, 14]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(248−13, 248, large)-Net over F₃ — Digital

Digital (235, 248, large)-net over F₃, using

3⁷ times duplication [i] based on digital (228, 241, large)-net over F₃, using
- t-expansion [i] based on digital (221, 241, large)-net over F₃, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3²⁴¹, large, F₃, 20) (dual of [large, large−241, 21]-code), using
    - strength reduction [i] based on linear OA(3²⁴¹, large, F₃, 25) (dual of [large, large−241, 26]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 3³⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(248−13, 248, large)-Net in Base 3 — Upper bound on s

There is no (235, 248, large)-net in base 3, because

11 times m-reduction [i] would yield (235, 237, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]

Best Known (248−13, 248, s)-Nets in Base 3

(248−13, 248, 6990521)-Net over F3 — Constructive and digital

(248−13, 248, large)-Net over F3 — Digital

(248−13, 248, large)-Net in Base 3 — Upper bound on s

(248−13, 248, 6990521)-Net over F₃ — Constructive and digital

(248−13, 248, large)-Net over F₃ — Digital