MinT - Best Known (12, 12+5, s)-Nets in Base 7

Best Known (12, 12+5, s)-Nets in Base 7

(12, 12+5, 2409)-Net over F₇ — Constructive and digital

Digital (12, 17, 2409)-net over F₇, using

net defined by OOA [i] based on linear OOA(7¹⁷, 2409, F₇, 6, 5) (dual of [(2409, 6), 14437, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(7¹⁷, 2410, F₇, 2, 5) (dual of [(2410, 2), 4803, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(7³, 57, F₇, 2, 2) (dual of [(57, 2), 111, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(7³, 57, F₇, 2) (dual of [57, 54, 3]-code), using
        Hamming code H(3,7) [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]

(12, 12+5, 4763)-Net over F₇ — Digital

Digital (12, 17, 4763)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7¹⁷, 4763, F₇, 5) (dual of [4763, 4746, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(7³, 57, F₇, 2) (dual of [57, 54, 3]-code), using
    - Hamming code H(3,7) [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]

(12, 12+5, 1358775)-Net in Base 7 — Upper bound on s

There is no (12, 17, 1358776)-net in base 7, because

1 times m-reduction [i] would yield (12, 16, 1358776)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 33 232940 690449 > 7¹⁶ [i]