MinT - Best Known (13, 13+5, s)-Nets in Base 7

Best Known (13, 13+5, s)-Nets in Base 7

(13, 13+5, 2752)-Net over F₇ — Constructive and digital

Digital (13, 18, 2752)-net over F₇, using

net defined by OOA [i] based on linear OOA(7¹⁸, 2752, F₇, 6, 5) (dual of [(2752, 6), 16494, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(7¹⁸, 2753, F₇, 2, 5) (dual of [(2753, 2), 5488, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(7⁴, 400, F₇, 2, 2) (dual of [(400, 2), 796, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(7⁴, 400, F₇, 2) (dual of [400, 396, 3]-code), using
        Hamming code H(4,7) [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]

(13, 13+5, 5106)-Net over F₇ — Digital

Digital (13, 18, 5106)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7¹⁸, 5106, F₇, 5) (dual of [5106, 5088, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(7⁴, 400, F₇, 2) (dual of [400, 396, 3]-code), using
    - Hamming code H(4,7) [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]

(13, 13+5, 3594984)-Net in Base 7 — Upper bound on s

There is no (13, 18, 3594985)-net in base 7, because

1 times m-reduction [i] would yield (13, 17, 3594985)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 232 630616 553601 > 7¹⁷ [i]