MinT - Best Known (12−4, 12, s)-Nets in Base 7

Best Known (12−4, 12, s)-Nets in Base 7

(12−4, 12, 2352)-Net over F₇ — Constructive and digital

Digital (8, 12, 2352)-net over F₇, using

net defined by OOA [i] based on linear OOA(7¹², 2352, F₇, 4, 4) (dual of [(2352, 4), 9396, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(7¹², 2352, F₇, 3, 4) (dual of [(2352, 3), 7044, 5]-NRT-code), using
  - OA 2-folding and stacking [i] based on linear OA(7¹², 4704, F₇, 4) (dual of [4704, 4692, 5]-code), using
    - trace code [i] based on linear OA(49⁶, 2352, F₄₉, 4) (dual of [2352, 2346, 5]-code), using
      - 1 times truncation [i] based on linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
        a code by Danev and Olsson [i]

(12−4, 12, 4704)-Net over F₇ — Digital

Digital (8, 12, 4704)-net over F₇, using

net defined by OOA [i] based on linear OOA(7¹², 4704, F₇, 4, 4) (dual of [(4704, 4), 18804, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(7¹², 4704, F₇, 3, 4) (dual of [(4704, 3), 14100, 5]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7¹², 4704, F₇, 4) (dual of [4704, 4692, 5]-code), using
    - trace code [i] based on linear OA(49⁶, 2352, F₄₉, 4) (dual of [2352, 2346, 5]-code), using
      - 1 times truncation [i] based on linear OA(49⁷, 2353, F₄₉, 5) (dual of [2353, 2346, 6]-code), using
        a code by Danev and Olsson [i]

(12−4, 12, 27729)-Net in Base 7 — Upper bound on s

There is no (8, 12, 27730)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 13841 984101 > 7¹² [i]