MinT - Best Known (19−5, 19, s)-Nets in Base 7

Best Known (19−5, 19, s)-Nets in Base 7

(19−5, 19, 4705)-Net over F₇ — Constructive and digital

Digital (14, 19, 4705)-net over F₇, using

net defined by OOA [i] based on linear OOA(7¹⁹, 4705, F₇, 6, 5) (dual of [(4705, 6), 28211, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(7¹⁹, 4706, F₇, 2, 5) (dual of [(4706, 2), 9393, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. 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]
    2. 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]

(19−5, 19, 7507)-Net over F₇ — Digital

Digital (14, 19, 7507)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7¹⁹, 7507, F₇, 5) (dual of [7507, 7488, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(7⁵, 2801, F₇, 2) (dual of [2801, 2796, 3]-code), using
    - Hamming code H(5,7) [i]
  2. 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]

(19−5, 19, large)-Net in Base 7 — Upper bound on s

There is no (14, 19, large)-net in base 7, because

3 times m-reduction [i] would yield (14, 16, large)-net in base 7, but
- mutually orthogonal hypercube bound [i]