MinT - Best Known (64−17, 64, s)-Nets in Base 81

Best Known (64−17, 64, s)-Nets in Base 81

(64−17, 64, 68070)-Net over F₈₁ — Constructive and digital

Digital (47, 64, 68070)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (7, 15, 1640)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81¹⁵, 1640, F₈₁, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(81¹⁵, 6560, F₈₁, 8) (dual of [6560, 6545, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(81¹⁵, 6561, F₈₁, 8) (dual of [6561, 6546, 9]-code), using
        an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 6560Â = 81²âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
2. digital (32, 49, 66430)-net over F₈₁, using
  - net defined by OOA [i] based on linear OOA(81⁴⁹, 66430, F₈₁, 17, 17) (dual of [(66430, 17), 1129261, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(81⁴⁹, 531441, F₈₁, 17) (dual of [531441, 531392, 18]-code), using
      - an extension C_e(16) of the primitive narrow-sense BCH-code C(I) with length 531440Â = 81³âˆ’1, defining interval IÂ =Â [1,16], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 17 [i]

(64−17, 64, 3659230)-Net over F₈₁ — Digital

Digital (47, 64, 3659230)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(64−17, 64, large)-Net in Base 81 — Upper bound on s

There is no (47, 64, large)-net in base 81, because

15 times m-reduction [i] would yield (47, 49, large)-net in base 81, but
- mutually orthogonal hypercube bound [i]