MinT - Best Known (4, 4+5, s)-Nets in Base 27

Best Known (4, 4+5, s)-Nets in Base 27

(4, 4+5, 379)-Net over F₂₇ — Constructive and digital

Digital (4, 9, 379)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (0, 2, 28)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
2. 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]

(4, 4+5, 458)-Net over F₂₇ — Digital

Digital (4, 9, 458)-net over F₂₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁹, 458, F₂₇, 5) (dual of [458, 449, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(27⁹, 728, F₂₇, 5) (dual of [728, 719, 6]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 728Â = 27²âˆ’1, defining interval IÂ =Â [0,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]

(4, 4+5, 28906)-Net in Base 27 — Upper bound on s

There is no (4, 9, 28907)-net in base 27, because

1 times m-reduction [i] would yield (4, 8, 28907)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 282449 025093 > 27⁸ [i]