MinT - Best Known (3, 3+5, s)-Nets in Base 27

Best Known (3, 3+5, s)-Nets in Base 27

(3, 3+5, 351)-Net over F₂₇ — Constructive and digital

Digital (3, 8, 351)-net over F₂₇, using

27¹ times duplication [i] based on digital (2, 7, 351)-net over F₂₇, using
- net defined by OOA [i] based on linear OOA(27⁷, 351, F₂₇, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
  - OOA 2-folding and stacking with additional row [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
    - a code by Danev and Olsson [i]

(3, 3+5, 352)-Net over F₂₇ — Digital

Digital (3, 8, 352)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(27⁸, 352, F₂₇, 2, 5) (dual of [(352, 2), 696, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(27⁸, 704, F₂₇, 5) (dual of [704, 696, 6]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(27⁷, 703, F₂₇, 5) (dual of [703, 696, 6]-code), using
    - a code by Danev and Olsson [i]

(3, 3+5, 5562)-Net in Base 27 — Upper bound on s

There is no (3, 8, 5563)-net in base 27, because

1 times m-reduction [i] would yield (3, 7, 5563)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 10462 245093 > 27⁷ [i]