Best Known (68, 68+19, s)-Nets in Base 32
(68, 68+19, 116585)-Net over F32 — Constructive and digital
Digital (68, 87, 116585)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (5, 14, 77)-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 (1, 10, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (0, 4, 33)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (54, 73, 116508)-net over F32, using
- net defined by OOA [i] based on linear OOA(3273, 116508, F32, 19, 19) (dual of [(116508, 19), 2213579, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3273, 1048573, F32, 19) (dual of [1048573, 1048500, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3273, 1048576, F32, 19) (dual of [1048576, 1048503, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(3273, 1048576, F32, 19) (dual of [1048576, 1048503, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3273, 1048573, F32, 19) (dual of [1048573, 1048500, 20]-code), using
- net defined by OOA [i] based on linear OOA(3273, 116508, F32, 19, 19) (dual of [(116508, 19), 2213579, 20]-NRT-code), using
- digital (5, 14, 77)-net over F32, using
(68, 68+19, 233061)-Net in Base 32 — Constructive
(68, 87, 233061)-net in base 32, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- (58, 77, 233017)-net in base 32, using
- base change [i] based on digital (36, 55, 233017)-net over F128, using
- net defined by OOA [i] based on linear OOA(12855, 233017, F128, 19, 19) (dual of [(233017, 19), 4427268, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12855, 2097154, F128, 19) (dual of [2097154, 2097099, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12855, 2097155, F128, 19) (dual of [2097155, 2097100, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 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(12855, 2097155, F128, 19) (dual of [2097155, 2097100, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12855, 2097154, F128, 19) (dual of [2097154, 2097099, 20]-code), using
- net defined by OOA [i] based on linear OOA(12855, 233017, F128, 19, 19) (dual of [(233017, 19), 4427268, 20]-NRT-code), using
- base change [i] based on digital (36, 55, 233017)-net over F128, using
- digital (1, 10, 44)-net over F32, using
(68, 68+19, 4588395)-Net over F32 — Digital
Digital (68, 87, 4588395)-net over F32, using
(68, 68+19, large)-Net in Base 32 — Upper bound on s
There is no (68, 87, large)-net in base 32, because
- 17 times m-reduction [i] would yield (68, 70, large)-net in base 32, but