MinT - Best Known (109−13, 109, s)-Nets in Base 27

Best Known (109−13, 109, s)-Nets in Base 27

(109−13, 109, 4214796)-Net over F₂₇ — Constructive and digital

Digital (96, 109, 4214796)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 6, 20496)-net over F₂₇, using
  - net defined by OOA [i] based on linear OOA(27⁶, 20496, F₂₇, 3, 3) (dual of [(20496, 3), 61482, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(27⁶, 20496, F₂₇, 2, 3) (dual of [(20496, 2), 40986, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(27⁶, 20496, F₂₇, 3) (dual of [20496, 20490, 4]-code or 20496-cap in PG(5,27)), using
        large caps in PG(5,Â b) [i]
2. digital (12, 16, 1398100)-net over F₂₇, using
  - s-reduction based on digital (12, 16, 4194301)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27¹⁶, 4194301, F₂₇, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
      - OA 2-folding and stacking [i] based on linear OA(27¹⁶, 8388602, F₂₇, 4) (dual of [8388602, 8388586, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹⁶, large, F₂₇, 4) (dual of [large, large−16, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
3. digital (20, 26, 1398100)-net over F₂₇, using
  - s-reduction based on digital (20, 26, 2796201)-net over F₂₇, using
    - net defined by OOA [i] based on linear OOA(27²⁶, 2796201, F₂₇, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
      - OA 3-folding and stacking [i] based on linear OA(27²⁶, large, F₂₇, 6) (dual of [large, large−26, 7]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
4. 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]

(109−13, 109, large)-Net over F₂₇ — Digital

Digital (96, 109, large)-net over F₂₇, using

27³ times duplication [i] based on digital (93, 106, large)-net over F₂₇, using
- t-expansion [i] based on digital (84, 106, large)-net over F₂₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27¹⁰⁶, large, F₂₇, 22) (dual of [large, large−106, 23]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(109−13, 109, large)-Net in Base 27 — Upper bound on s

There is no (96, 109, large)-net in base 27, because

11 times m-reduction [i] would yield (96, 98, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (109−13, 109, s)-Nets in Base 27

(109−13, 109, 4214796)-Net over F27 — Constructive and digital

(109−13, 109, large)-Net over F27 — Digital

(109−13, 109, large)-Net in Base 27 — Upper bound on s

(109−13, 109, 4214796)-Net over F₂₇ — Constructive and digital

(109−13, 109, large)-Net over F₂₇ — Digital