Best Known (11, 11+8, s)-Nets in Base 32
(11, 11+8, 289)-Net over F32 — Constructive and digital
Digital (11, 19, 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 (7, 15, 256)-net over F32, using
- net defined by OOA [i] based on linear OOA(3215, 256, F32, 8, 8) (dual of [(256, 8), 2033, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(3215, 1024, F32, 8) (dual of [1024, 1009, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OA 4-folding and stacking [i] based on linear OA(3215, 1024, F32, 8) (dual of [1024, 1009, 9]-code), using
- net defined by OOA [i] based on linear OOA(3215, 256, F32, 8, 8) (dual of [(256, 8), 2033, 9]-NRT-code), using
- digital (0, 4, 33)-net over F32, using
(11, 11+8, 1024)-Net in Base 32 — Constructive
(11, 19, 1024)-net in base 32, using
- 321 times duplication [i] based on (10, 18, 1024)-net in base 32, using
- base change [i] based on digital (7, 15, 1024)-net over F64, using
- net defined by OOA [i] based on linear OOA(6415, 1024, F64, 8, 8) (dual of [(1024, 8), 8177, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on 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]
- OA 4-folding and stacking [i] based on linear OA(6415, 4096, F64, 8) (dual of [4096, 4081, 9]-code), using
- net defined by OOA [i] based on linear OOA(6415, 1024, F64, 8, 8) (dual of [(1024, 8), 8177, 9]-NRT-code), using
- base change [i] based on digital (7, 15, 1024)-net over F64, using
(11, 11+8, 1422)-Net over F32 — Digital
Digital (11, 19, 1422)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3219, 1422, F32, 8) (dual of [1422, 1403, 9]-code), using
- 391 step Varšamov–Edel lengthening with (ri) = (2, 11 times 0, 1, 72 times 0, 1, 305 times 0) [i] based on linear OA(3215, 1027, F32, 8) (dual of [1027, 1012, 9]-code), using
- construction XX applied to C1 = C([1022,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([1022,6]) [i] based on
- linear OA(3213, 1023, F32, 7) (dual of [1023, 1010, 8]-code), using the primitive BCH-code C(I) with length 1023 = 322−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [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(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(3211, 1023, F32, 6) (dual of [1023, 1012, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 322−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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([1022,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([1022,6]) [i] based on
- 391 step Varšamov–Edel lengthening with (ri) = (2, 11 times 0, 1, 72 times 0, 1, 305 times 0) [i] based on linear OA(3215, 1027, F32, 8) (dual of [1027, 1012, 9]-code), using
(11, 11+8, 2049)-Net in Base 32
(11, 19, 2049)-net in base 32, using
- 321 times duplication [i] based on (10, 18, 2049)-net in base 32, using
- base change [i] based on digital (7, 15, 2049)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6415, 2049, F64, 2, 8) (dual of [(2049, 2), 4083, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(6415, 4098, F64, 8) (dual of [4098, 4083, 9]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6415, 2049, F64, 2, 8) (dual of [(2049, 2), 4083, 9]-NRT-code), using
- base change [i] based on digital (7, 15, 2049)-net over F64, using
(11, 11+8, 1007286)-Net in Base 32 — Upper bound on s
There is no (11, 19, 1007287)-net in base 32, because
- the generalized Rao bound for nets shows that 32m ≥ 39614 233677 270712 084404 147323 > 3219 [i]