MinT - Best Known (7, 7+4, s)-Nets in Base 8

Best Known (7, 7+4, s)-Nets in Base 8

(7, 7+4, 2016)-Net over F₈ — Constructive and digital

Digital (7, 11, 2016)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹¹, 2016, F₈, 4, 4) (dual of [(2016, 4), 8053, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(8¹¹, 2016, F₈, 3, 4) (dual of [(2016, 3), 6037, 5]-NRT-code), using
  - OA 2-folding and stacking [i] based on linear OA(8¹¹, 4032, F₈, 4) (dual of [4032, 4021, 5]-code), using
    - base reduction for projective spaces (embedding PG(5,64) in PG(10,8)) [i] based on linear OA(64⁶, 4032, F₆₄, 4) (dual of [4032, 4026, 5]-code), using
      - 1 times truncation [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]

(7, 7+4, 4032)-Net over F₈ — Digital

Digital (7, 11, 4032)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹¹, 4032, F₈, 4, 4) (dual of [(4032, 4), 16117, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(8¹¹, 4032, F₈, 3, 4) (dual of [(4032, 3), 12085, 5]-NRT-code), using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹, 4032, F₈, 4) (dual of [4032, 4021, 5]-code), using
    - base reduction for projective spaces (embedding PG(5,64) in PG(10,8)) [i] based on linear OA(64⁶, 4032, F₆₄, 4) (dual of [4032, 4026, 5]-code), using
      - 1 times truncation [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]

(7, 7+4, 18723)-Net in Base 8 — Upper bound on s

There is no (7, 11, 18724)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8590 131187 > 8¹¹ [i]