Best Known (109−32, 109, s)-Nets in Base 32
(109−32, 109, 2080)-Net over F32 — Constructive and digital
Digital (77, 109, 2080)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 16, 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 (61, 93, 2047)-net over F32, using
- net defined by OOA [i] based on linear OOA(3293, 2047, F32, 32, 32) (dual of [(2047, 32), 65411, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(3293, 32752, F32, 32) (dual of [32752, 32659, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3293, 32767, F32, 32) (dual of [32767, 32674, 33]-code), using
- 1 times truncation [i] based on linear OA(3294, 32768, F32, 33) (dual of [32768, 32674, 34]-code), using
- an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- 1 times truncation [i] based on linear OA(3294, 32768, F32, 33) (dual of [32768, 32674, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3293, 32767, F32, 32) (dual of [32767, 32674, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(3293, 32752, F32, 32) (dual of [32752, 32659, 33]-code), using
- net defined by OOA [i] based on linear OOA(3293, 2047, F32, 32, 32) (dual of [(2047, 32), 65411, 33]-NRT-code), using
- digital (0, 16, 33)-net over F32, using
(109−32, 109, 4097)-Net in Base 32 — Constructive
(77, 109, 4097)-net in base 32, using
- net defined by OOA [i] based on OOA(32109, 4097, S32, 32, 32), using
- OA 16-folding and stacking [i] based on OA(32109, 65552, S32, 32), using
- discarding factors based on OA(32109, 65553, S32, 32), using
- discarding parts of the base [i] based on linear OA(25668, 65553, F256, 32) (dual of [65553, 65485, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(25) [i] based on
- linear OA(25663, 65536, F256, 32) (dual of [65536, 65473, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(25651, 65536, F256, 26) (dual of [65536, 65485, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2565, 17, F256, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,256)), using
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- Reed–Solomon code RS(251,256) [i]
- discarding factors / shortening the dual code based on linear OA(2565, 256, F256, 5) (dual of [256, 251, 6]-code or 256-arc in PG(4,256)), using
- construction X applied to Ce(31) ⊂ Ce(25) [i] based on
- discarding parts of the base [i] based on linear OA(25668, 65553, F256, 32) (dual of [65553, 65485, 33]-code), using
- discarding factors based on OA(32109, 65553, S32, 32), using
- OA 16-folding and stacking [i] based on OA(32109, 65552, S32, 32), using
(109−32, 109, 78534)-Net over F32 — Digital
Digital (77, 109, 78534)-net over F32, using
(109−32, 109, large)-Net in Base 32 — Upper bound on s
There is no (77, 109, large)-net in base 32, because
- 30 times m-reduction [i] would yield (77, 79, large)-net in base 32, but