Best Known (21−9, 21, s)-Nets in Base 32
(21−9, 21, 289)-Net over F32 — Constructive and digital
Digital (12, 21, 289)-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 (8, 17, 256)-net over F32, using
- net defined by OOA [i] based on linear OOA(3217, 256, F32, 9, 9) (dual of [(256, 9), 2287, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(3217, 1025, F32, 9) (dual of [1025, 1008, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 324−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(3217, 1025, F32, 9) (dual of [1025, 1008, 10]-code), using
- net defined by OOA [i] based on linear OOA(3217, 256, F32, 9, 9) (dual of [(256, 9), 2287, 10]-NRT-code), using
- digital (0, 4, 33)-net over F32, using
(21−9, 21, 1024)-Net in Base 32 — Constructive
(12, 21, 1024)-net in base 32, using
- net defined by OOA [i] based on OOA(3221, 1024, S32, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(3221, 4097, S32, 9), using
- discarding factors based on OA(3221, 4098, S32, 9), using
- discarding parts of the base [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- discarding parts of the base [i] based on linear OA(6417, 4098, F64, 9) (dual of [4098, 4081, 10]-code), using
- discarding factors based on OA(3221, 4098, S32, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(3221, 4097, S32, 9), using
(21−9, 21, 1227)-Net over F32 — Digital
Digital (12, 21, 1227)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3221, 1227, F32, 9) (dual of [1227, 1206, 10]-code), using
- 196 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 times 0, 1, 156 times 0) [i] based on linear OA(3217, 1027, F32, 9) (dual of [1027, 1010, 10]-code), using
- construction XX applied to C1 = C([1022,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1022,7]) [i] based on
- linear OA(3215, 1023, F32, 8) (dual of [1023, 1008, 9]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3215, 1023, F32, 8) (dual of [1023, 1008, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(3217, 1023, F32, 9) (dual of [1023, 1006, 10]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(3213, 1023, F32, 7) (dual of [1023, 1010, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(320, 2, F32, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([1022,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([1022,7]) [i] based on
- 196 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 32 times 0, 1, 156 times 0) [i] based on linear OA(3217, 1027, F32, 9) (dual of [1027, 1010, 10]-code), using
(21−9, 21, 2395745)-Net in Base 32 — Upper bound on s
There is no (12, 21, 2395746)-net in base 32, because
- 1 times m-reduction [i] would yield (12, 20, 2395746)-net in base 32, but
- the generalized Rao bound for nets shows that 32m ≥ 1 267651 287037 257134 102066 686216 > 3220 [i]