MinT - Best Known (7, 12, s)-Nets in Base 27

Best Known (7, 12, s)-Nets in Base 27

(7, 12, 1405)-Net over F₂₇ — Constructive and digital

Digital (7, 12, 1405)-net over F₂₇, using

net defined by OOA [i] based on linear OOA(27¹², 1405, F₂₇, 5, 5) (dual of [(1405, 5), 7013, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(27¹², 2811, F₂₇, 5) (dual of [2811, 2799, 6]-code), using
  - discarding factors / shortening the dual code based on linear OA(27¹², 2812, F₂₇, 5) (dual of [2812, 2800, 6]-code), using
    - generalized (u,Â u+v)-construction [i] based on
      1. linear OA(27¹, 703, F₂₇, 1) (dual of [703, 702, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
      2. linear OA(27¹, 703, F₂₇, 1) (dual of [703, 702, 2]-code) (see above)
      3. linear OA(27³, 703, F₂₇, 2) (dual of [703, 700, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(27³, 728, F₂₇, 2) (dual of [728, 725, 3]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
      4. linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
        a code by Danev and Olsson [i]

(7, 12, 2812)-Net over F₂₇ — Digital

Digital (7, 12, 2812)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27¹², 2812, F₂₇, 5) (dual of [2812, 2800, 6]-code), using
- generalized (u,Â u+v)-construction [i] based on
  1. linear OA(27¹, 703, F₂₇, 1) (dual of [703, 702, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(27¹, s, F₂₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]
  2. linear OA(27¹, 703, F₂₇, 1) (dual of [703, 702, 2]-code) (see above)
  3. linear OA(27³, 703, F₂₇, 2) (dual of [703, 700, 3]-code), using
    - discarding factors / shortening the dual code based on linear OA(27³, 728, F₂₇, 2) (dual of [728, 725, 3]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
  4. linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
    - a code by Danev and Olsson [i]

(7, 12, 3322)-Net in Base 27 — Constructive

(7, 12, 3322)-net in base 27, using

base change [i] based on digital (4, 9, 3322)-net over F₈₁, using
- net defined by OOA [i] based on linear OOA(81⁹, 3322, F₈₁, 5, 5) (dual of [(3322, 5), 16601, 6]-NRT-code), using
  - appending kth column [i] based on linear OOA(81⁹, 3322, F₈₁, 4, 5) (dual of [(3322, 4), 13279, 6]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(81², 82, F₈₁, 4, 2) (dual of [(82, 4), 326, 3]-NRT-code), using
        extended Reedâ€“Solomon NRT-code RS_e(4;326,81) [i]
      2. linear OOA(81⁷, 3240, F₈₁, 4, 5) (dual of [(3240, 4), 12953, 6]-NRT-code), using
        OOA 2-folding and stacking with additional row [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(7, 12, 4055480)-Net in Base 27 — Upper bound on s

There is no (7, 12, 4055481)-net in base 27, because

1 times m-reduction [i] would yield (7, 11, 4055481)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 5559 062617 417609 > 27¹¹ [i]