Best Known (32, 32+8, s)-Nets in Base 32
(32, 32+8, 2097183)-Net over F32 — Constructive and digital
Digital (32, 40, 2097183)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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
- digital (28, 36, 2097150)-net over F32, using
- net defined by OOA [i] based on linear OOA(3236, 2097150, F32, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(3236, 8388600, F32, 8) (dual of [8388600, 8388564, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(3236, large, F32, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(3236, large, F32, 8) (dual of [large, large−36, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(3236, 8388600, F32, 8) (dual of [8388600, 8388564, 9]-code), using
- net defined by OOA [i] based on linear OOA(3236, 2097150, F32, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- digital (0, 4, 33)-net over F32, using
(32, 32+8, 2097215)-Net in Base 32 — Constructive
(32, 40, 2097215)-net in base 32, using
- (u, u+v)-construction [i] based on
- (1, 5, 65)-net in base 32, using
- 1 times m-reduction [i] based on (1, 6, 65)-net in base 32, using
- base change [i] based on digital (0, 5, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- base change [i] based on digital (0, 5, 65)-net over F64, using
- 1 times m-reduction [i] based on (1, 6, 65)-net in base 32, using
- (27, 35, 2097150)-net in base 32, using
- net defined by OOA [i] based on OOA(3235, 2097150, S32, 8, 8), using
- OA 4-folding and stacking [i] based on OA(3235, 8388600, S32, 8), using
- discarding factors based on OA(3235, large, S32, 8), using
- discarding parts of the base [i] based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding parts of the base [i] based on linear OA(6429, large, F64, 8) (dual of [large, large−29, 9]-code), using
- discarding factors based on OA(3235, large, S32, 8), using
- OA 4-folding and stacking [i] based on OA(3235, 8388600, S32, 8), using
- net defined by OOA [i] based on OOA(3235, 2097150, S32, 8, 8), using
- (1, 5, 65)-net in base 32, using
(32, 32+8, large)-Net over F32 — Digital
Digital (32, 40, large)-net over F32, using
- 1 times m-reduction [i] based on digital (32, 41, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3241, large, F32, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3241, large, F32, 9) (dual of [large, large−41, 10]-code), using
(32, 32+8, large)-Net in Base 32 — Upper bound on s
There is no (32, 40, large)-net in base 32, because
- 6 times m-reduction [i] would yield (32, 34, large)-net in base 32, but