Best Known (44, 57, s)-Nets in Base 256
(44, 57, 2796457)-Net over F256 — Constructive and digital
Digital (44, 57, 2796457)-net over F256, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 4, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- digital (10, 16, 1398100)-net over F256, using
- s-reduction based on digital (10, 16, 2796201)-net over F256, using
- net defined by OOA [i] based on linear OOA(25616, 2796201, F256, 6, 6) (dual of [(2796201, 6), 16777190, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- net defined by OOA [i] based on linear OOA(25616, 2796201, F256, 6, 6) (dual of [(2796201, 6), 16777190, 7]-NRT-code), using
- s-reduction based on digital (10, 16, 2796201)-net over F256, using
- digital (24, 37, 1398100)-net over F256, using
- net defined by OOA [i] based on linear OOA(25637, 1398100, F256, 13, 13) (dual of [(1398100, 13), 18175263, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(25637, 8388601, F256, 13) (dual of [8388601, 8388564, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(25637, 8388601, F256, 13) (dual of [8388601, 8388564, 14]-code), using
- net defined by OOA [i] based on linear OOA(25637, 1398100, F256, 13, 13) (dual of [(1398100, 13), 18175263, 14]-NRT-code), using
- digital (0, 4, 257)-net over F256, using
(44, 57, large)-Net over F256 — Digital
Digital (44, 57, large)-net over F256, using
- t-expansion [i] based on digital (42, 57, large)-net over F256, using
- 4 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
- 4 times m-reduction [i] based on digital (42, 61, large)-net over F256, using
(44, 57, large)-Net in Base 256 — Upper bound on s
There is no (44, 57, large)-net in base 256, because
- 11 times m-reduction [i] would yield (44, 46, large)-net in base 256, but