MinT - Best Known (21−5, 21, s)-Nets in Base 7

Best Known (21−5, 21, s)-Nets in Base 7

(21−5, 21, 9411)-Net over F₇ — Constructive and digital

Digital (16, 21, 9411)-net over F₇, using

net defined by OOA [i] based on linear OOA(7²¹, 9411, F₇, 6, 5) (dual of [(9411, 6), 56445, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(7²¹, 9412, F₇, 2, 5) (dual of [(9412, 2), 18803, 6]-NRT-code), using
  - generalized (u,Â u+v)-construction [i] based on
    1. linear OOA(7¹, 2353, F₇, 2, 1) (dual of [(2353, 2), 4705, 2]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(7¹, s, F₇, 2, 1) with arbitrarily large s, using
        appending 1 arbitrary column [i] based on linear OA(7¹, s, F₇, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
    2. linear OOA(7¹, 2353, F₇, 2, 1) (dual of [(2353, 2), 4705, 2]-NRT-code) (see above)
    3. linear OOA(7⁵, 2353, F₇, 2, 2) (dual of [(2353, 2), 4701, 3]-NRT-code), using
      - discarding factors / shortening the dual code based on linear OOA(7⁵, 2801, F₇, 2, 2) (dual of [(2801, 2), 5597, 3]-NRT-code), using
        appending kth column [i] based on linear OA(7⁵, 2801, F₇, 2) (dual of [2801, 2796, 3]-code), using
        Hamming code H(5,7) [i]
    4. linear OOA(7¹⁴, 2353, F₇, 2, 5) (dual of [(2353, 2), 4692, 6]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(7¹⁴, 4706, F₇, 5) (dual of [4706, 4692, 6]-code), using
        trace code [i] based on linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
        a code by Danev and Olsson [i]

(21−5, 21, 16812)-Net over F₇ — Digital

Digital (16, 21, 16812)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7²¹, 16812, F₇, 5) (dual of [16812, 16791, 6]-code), using
- constructionÂ X applied to C_e(4) âŠ‚ C_e(3) [i] based on
  1. linear OA(7²¹, 16807, F₇, 5) (dual of [16807, 16786, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 16806Â = 7⁵âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
  2. linear OA(7¹⁶, 16807, F₇, 4) (dual of [16807, 16791, 5]-code), using an extension C_e(3) of the primitive narrow-sense BCH-code C(I) with length 16806Â = 7⁵âˆ’1, defining interval IÂ =Â [1,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 4 [i]
  3. linear OA(7⁰, 5, F₇, 0) (dual of [5, 5, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(7⁰, s, F₇, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(21−5, 21, large)-Net in Base 7 — Upper bound on s

There is no (16, 21, large)-net in base 7, because

3 times m-reduction [i] would yield (16, 18, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]