Best Known (50−10, 50, s)-Nets in Base 32
(50−10, 50, 1677720)-Net over F32 — Constructive and digital
Digital (40, 50, 1677720)-net over F32, using
- 1 times m-reduction [i] based on digital (40, 51, 1677720)-net over F32, using
- net defined by OOA [i] based on linear OOA(3251, 1677720, F32, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3251, 8388601, F32, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3251, large, F32, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3251, large, F32, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3251, 8388601, F32, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(3251, 1677720, F32, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
(50−10, 50, 1677753)-Net in Base 32 — Constructive
(40, 50, 1677753)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 33)-net over F32, using
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 0 and N(F) ≥ 33, using
- the rational function field F32(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 32)-sequence over F32, using
- (35, 45, 1677720)-net in base 32, using
- net defined by OOA [i] based on OOA(3245, 1677720, S32, 10, 10), using
- OA 5-folding and stacking [i] based on OA(3245, 8388600, S32, 10), using
- discarding factors based on OA(3245, large, S32, 10), using
- discarding parts of the base [i] based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding parts of the base [i] based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- discarding factors based on OA(3245, large, S32, 10), using
- OA 5-folding and stacking [i] based on OA(3245, 8388600, S32, 10), using
- net defined by OOA [i] based on OOA(3245, 1677720, S32, 10, 10), using
- digital (0, 5, 33)-net over F32, using
(50−10, 50, large)-Net over F32 — Digital
Digital (40, 50, large)-net over F32, using
- 1 times m-reduction [i] based on digital (40, 51, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3251, large, F32, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3251, large, F32, 11) (dual of [large, large−51, 12]-code), using
(50−10, 50, large)-Net in Base 32 — Upper bound on s
There is no (40, 50, large)-net in base 32, because
- 8 times m-reduction [i] would yield (40, 42, large)-net in base 32, but