MinT - Best Known (9−5, 9, s)-Nets in Base 81

Best Known (9−5, 9, s)-Nets in Base 81

(9−5, 9, 3322)-Net over F₈₁ — Constructive and digital

Digital (4, 9, 3322)-net over F₈₁, using

net defined by OOA [i] based on linear OOA(81⁹, 3322, F₈₁, 5, 5) (dual of [(3322, 5), 16601, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(81⁹, 3322, F₈₁, 4, 5) (dual of [(3322, 4), 13279, 6]-NRT-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OOA(81², 82, F₈₁, 4, 2) (dual of [(82, 4), 326, 3]-NRT-code), using
      - extended Reedâ€“Solomon NRT-code RS_e(4;326,81) [i]
    2. linear OOA(81⁷, 3240, F₈₁, 4, 5) (dual of [(3240, 4), 12953, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(81⁷, 6481, F₈₁, 5) (dual of [6481, 6474, 6]-code), using
        a code by Danev and Olsson [i]

(9−5, 9, 760965)-Net in Base 81 — Upper bound on s

There is no (4, 9, 760966)-net in base 81, because

1 times m-reduction [i] would yield (4, 8, 760966)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 1853 024166 944961 > 81⁸ [i]