MinT - Best Known (59−9, 59, s)-Nets in Base 27

Best Known (59−9, 59, s)-Nets in Base 27

(59−9, 59, 4194301)-Net over F₂₇ — Constructive and digital

Digital (50, 59, 4194301)-net over F₂₇, using

net defined by OOA [i] based on linear OOA(27⁵⁹, 4194301, F₂₇, 10, 9) (dual of [(4194301, 10), 41942951, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(27⁵⁹, large, F₂₇, 2, 9), using
  - 1 times NRT-code embedding in larger space [i] based on linear OOA(27⁵⁷, 8388602, F₂₇, 2, 9) (dual of [(8388602, 2), 16777147, 10]-NRT-code), using
    - (u,Â u+v)-construction [i] based on
      1. linear OOA(27¹⁶, 4194301, F₂₇, 2, 4) (dual of [(4194301, 2), 8388586, 5]-NRT-code), using
        OOA 2-folding [i] based on linear OA(27¹⁶, 8388602, F₂₇, 4) (dual of [8388602, 8388586, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(27¹⁶, large, F₂₇, 4) (dual of [large, large−16, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 14348906Â = 27⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
      2. linear OOA(27⁴¹, 4194301, F₂₇, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
        OOA 2-folding [i] based on linear OA(27⁴¹, 8388602, F₂₇, 9) (dual of [8388602, 8388561, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(27⁴¹, large, F₂₇, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(59−9, 59, large)-Net over F₂₇ — Digital

Digital (50, 59, large)-net over F₂₇, using

t-expansion [i] based on digital (48, 59, large)-net over F₂₇, using
- 2 times m-reduction [i] based on digital (48, 61, large)-net over F₂₇, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(27⁶¹, large, F₂₇, 13) (dual of [large, large−61, 14]-code), using
    - the expurgated narrow-sense BCH-code C(I) with length 14348908Â | 27¹⁰âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]

(59−9, 59, large)-Net in Base 27 — Upper bound on s

There is no (50, 59, large)-net in base 27, because

7 times m-reduction [i] would yield (50, 52, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]

Best Known (59−9, 59, s)-Nets in Base 27

(59−9, 59, 4194301)-Net over F27 — Constructive and digital

(59−9, 59, large)-Net over F27 — Digital

(59−9, 59, large)-Net in Base 27 — Upper bound on s

(59−9, 59, 4194301)-Net over F₂₇ — Constructive and digital

(59−9, 59, large)-Net over F₂₇ — Digital