MinT - Best Known (12−5, 12, s)-Nets in Base 25

Best Known (12−5, 12, s)-Nets in Base 25

(12−5, 12, 1201)-Net over F₂₅ — Constructive and digital

Digital (7, 12, 1201)-net over F₂₅, using

net defined by OOA [i] based on linear OOA(25¹², 1201, F₂₅, 5, 5) (dual of [(1201, 5), 5993, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(25¹², 2403, F₂₅, 5) (dual of [2403, 2391, 6]-code), using
  - discarding factors / shortening the dual code based on linear OA(25¹², 2404, F₂₅, 5) (dual of [2404, 2392, 6]-code), using
    - generalized (u,Â u+v)-construction [i] based on
      1. linear OA(25¹, 601, F₂₅, 1) (dual of [601, 600, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(25¹, s, F₂₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
      2. linear OA(25¹, 601, F₂₅, 1) (dual of [601, 600, 2]-code) (see above)
      3. linear OA(25³, 601, F₂₅, 2) (dual of [601, 598, 3]-code), using
        discarding factors / shortening the dual code based on linear OA(25³, 624, F₂₅, 2) (dual of [624, 621, 3]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
      4. linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
        a code by Danev and Olsson [i]

(12−5, 12, 2404)-Net over F₂₅ — Digital

Digital (7, 12, 2404)-net over F₂₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(25¹², 2404, F₂₅, 5) (dual of [2404, 2392, 6]-code), using
- generalized (u,Â u+v)-construction [i] based on
  1. linear OA(25¹, 601, F₂₅, 1) (dual of [601, 600, 2]-code), using
    - discarding factors / shortening the dual code based on linear OA(25¹, s, F₂₅, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
      - parity-check code [i]
  2. linear OA(25¹, 601, F₂₅, 1) (dual of [601, 600, 2]-code) (see above)
  3. linear OA(25³, 601, F₂₅, 2) (dual of [601, 598, 3]-code), using
    - discarding factors / shortening the dual code based on linear OA(25³, 624, F₂₅, 2) (dual of [624, 621, 3]-code), using
      - the primitive expurgated narrow-sense BCH-code C(I) with length 624Â = 25²âˆ’1, defining interval IÂ =Â [0,1], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 3 [i]
  4. linear OA(25⁷, 601, F₂₅, 5) (dual of [601, 594, 6]-code), using
    - a code by Danev and Olsson [i]

(12−5, 12, 2877224)-Net in Base 25 — Upper bound on s

There is no (7, 12, 2877225)-net in base 25, because

1 times m-reduction [i] would yield (7, 11, 2877225)-net in base 25, but
- the generalized Rao bound for nets shows that 25^mÂ â‰¥ 2384 186992 527601 > 25¹¹ [i]