Best Known (43, 43+15, s)-Nets in Base 256
(43, 43+15, 1220218)-Net over F256 — Constructive and digital
Digital (43, 58, 1220218)-net over F256, using
- 2561 times duplication [i] based on digital (42, 57, 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 (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- digital (7, 14, 21847)-net over F256, using
- (u, u+v)-construction [i] based on
(43, 43+15, large)-Net over F256 — Digital
Digital (43, 58, large)-net over F256, using
- t-expansion [i] based on digital (42, 58, large)-net over F256, using
- 3 times m-reduction [i] based on digital (42, 61, large)-net over F256, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25661, large, F256, 19) (dual of [large, large−61, 20]-code), using
- strength reduction [i] based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- strength reduction [i] based on linear OA(25661, large, F256, 21) (dual of [large, large−61, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25661, large, F256, 19) (dual of [large, large−61, 20]-code), using
- 3 times m-reduction [i] based on digital (42, 61, large)-net over F256, using
(43, 43+15, large)-Net in Base 256 — Upper bound on s
There is no (43, 58, large)-net in base 256, because
- 13 times m-reduction [i] would yield (43, 45, large)-net in base 256, but