MinT - Best Known (12, 17, s)-Nets in Base 8

Best Known (12, 17, s)-Nets in Base 8

(12, 17, 4105)-Net over F₈ — Constructive and digital

Digital (12, 17, 4105)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹⁷, 4105, F₈, 6, 5) (dual of [(4105, 6), 24613, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8¹⁷, 4106, F₈, 2, 5) (dual of [(4106, 2), 8195, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(8³, 73, F₈, 2, 2) (dual of [(73, 2), 143, 3]-NRT-code), using
      - appending kth column [i] based on linear OA(8³, 73, F₈, 2) (dual of [73, 70, 3]-code), using
        Hamming code H(3,8) [i]
    2. linear OOA(8¹⁴, 4033, F₈, 2, 5) (dual of [(4033, 2), 8052, 6]-NRT-code), using
      - OOA 2-folding [i] based on linear OA(8¹⁴, 8066, F₈, 5) (dual of [8066, 8052, 6]-code), using
        trace code [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]

(12, 17, 8128)-Net in Base 8 — Constructive

(12, 17, 8128)-net in base 8, using

net defined by OOA [i] based on OOA(8¹⁷, 8128, S₈, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(8¹⁷, 16257, S₈, 5), using
  - discarding parts of the base [i] based on linear OA(128⁷, 16257, F₁₂₈, 5) (dual of [16257, 16250, 6]-code), using
    - a code by Danev and Olsson [i]

(12, 17, 8139)-Net over F₈ — Digital

Digital (12, 17, 8139)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁷, 8139, F₈, 5) (dual of [8139, 8122, 6]-code), using
- (u,Â u+v)-construction [i] based on
  1. linear OA(8³, 73, F₈, 2) (dual of [73, 70, 3]-code), using
    - Hamming code H(3,8) [i]
  2. linear OA(8¹⁴, 8066, F₈, 5) (dual of [8066, 8052, 6]-code), using
    - trace code [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
      - a code by Danev and Olsson [i]

(12, 17, 3389508)-Net in Base 8 — Upper bound on s

There is no (12, 17, 3389509)-net in base 8, because

1 times m-reduction [i] would yield (12, 16, 3389509)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 281 475026 392582 > 8¹⁶ [i]