Best Known (40, 40+14, s)-Nets in Base 256
(40, 40+14, 1220218)-Net over F256 — Constructive and digital
Digital (40, 54, 1220218)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (7, 14, 21847)-net over F256, using
- net defined by OOA [i] based on linear OOA(25614, 21847, F256, 7, 7) (dual of [(21847, 7), 152915, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(25614, 21847, F256, 6, 7) (dual of [(21847, 6), 131068, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(25613, 65537, F256, 7) (dual of [65537, 65524, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(2569, 65537, F256, 5) (dual of [65537, 65528, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(25614, 65542, F256, 7) (dual of [65542, 65528, 8]-code), using
- appending kth column [i] based on linear OOA(25614, 21847, F256, 6, 7) (dual of [(21847, 6), 131068, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25614, 21847, F256, 7, 7) (dual of [(21847, 7), 152915, 8]-NRT-code), using
- digital (26, 40, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- digital (7, 14, 21847)-net over F256, using
(40, 40+14, large)-Net over F256 — Digital
Digital (40, 54, large)-net over F256, using
- 4 times m-reduction [i] based on digital (40, 58, large)-net over F256, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25658, large, F256, 18) (dual of [large, large−58, 19]-code), using
- strength reduction [i] based on linear OA(25658, large, F256, 20) (dual of [large, large−58, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- strength reduction [i] based on linear OA(25658, large, F256, 20) (dual of [large, large−58, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25658, large, F256, 18) (dual of [large, large−58, 19]-code), using
(40, 40+14, large)-Net in Base 256 — Upper bound on s
There is no (40, 54, large)-net in base 256, because
- 12 times m-reduction [i] would yield (40, 42, large)-net in base 256, but