Best Known (31, 31+8, s)-Nets in Base 32
(31, 31+8, 2097150)-Net over F32 — Constructive and digital
Digital (31, 39, 2097150)-net over F32, using
- 323 times duplication [i] based on 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
(31, 31+8, 2097183)-Net in Base 32 — Constructive
(31, 39, 2097183)-net in base 32, 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
- (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
- digital (0, 4, 33)-net over F32, using
(31, 31+8, large)-Net over F32 — Digital
Digital (31, 39, large)-net over F32, using
- 323 times duplication [i] based on digital (28, 36, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3236, large, F32, 8) (dual of [large, large−36, 9]-code), using
(31, 31+8, large)-Net in Base 32 — Upper bound on s
There is no (31, 39, large)-net in base 32, because
- 6 times m-reduction [i] would yield (31, 33, large)-net in base 32, but