Best Known (45, 45+19, s)-Nets in Base 32
(45, 45+19, 3674)-Net over F32 — Constructive and digital
Digital (45, 64, 3674)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 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 (36, 55, 3641)-net over F32, using
- net defined by OOA [i] based on linear OOA(3255, 3641, F32, 19, 19) (dual of [(3641, 19), 69124, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3255, 32770, F32, 19) (dual of [32770, 32715, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3255, 32771, F32, 19) (dual of [32771, 32716, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(3255, 32768, F32, 19) (dual of [32768, 32713, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3252, 32768, F32, 18) (dual of [32768, 32716, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(320, 3, F32, 0) (dual of [3, 3, 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
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(3255, 32771, F32, 19) (dual of [32771, 32716, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3255, 32770, F32, 19) (dual of [32770, 32715, 20]-code), using
- net defined by OOA [i] based on linear OOA(3255, 3641, F32, 19, 19) (dual of [(3641, 19), 69124, 20]-NRT-code), using
- digital (0, 9, 33)-net over F32, using
(45, 45+19, 7283)-Net in Base 32 — Constructive
(45, 64, 7283)-net in base 32, using
- base change [i] based on digital (21, 40, 7283)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 7283, F256, 19, 19) (dual of [(7283, 19), 138337, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(25629, 65537, F256, 15) (dual of [65537, 65508, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(25640, 65548, F256, 19) (dual of [65548, 65508, 20]-code), using
- net defined by OOA [i] based on linear OOA(25640, 7283, F256, 19, 19) (dual of [(7283, 19), 138337, 20]-NRT-code), using
(45, 45+19, 54763)-Net over F32 — Digital
Digital (45, 64, 54763)-net over F32, using
(45, 45+19, large)-Net in Base 32 — Upper bound on s
There is no (45, 64, large)-net in base 32, because
- 17 times m-reduction [i] would yield (45, 47, large)-net in base 32, but