Best Known (61−18, 61, s)-Nets in Base 64
(61−18, 61, 29192)-Net over F64 — Constructive and digital
Digital (43, 61, 29192)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (34, 52, 29127)-net over F64, using
- net defined by OOA [i] based on linear OOA(6452, 29127, F64, 18, 18) (dual of [(29127, 18), 524234, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6452, 262143, F64, 18) (dual of [262143, 262091, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(6452, 262144, F64, 18) (dual of [262144, 262092, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(6452, 262144, F64, 18) (dual of [262144, 262092, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(6452, 262143, F64, 18) (dual of [262143, 262091, 19]-code), using
- net defined by OOA [i] based on linear OOA(6452, 29127, F64, 18, 18) (dual of [(29127, 18), 524234, 19]-NRT-code), using
- digital (0, 9, 65)-net over F64, using
(61−18, 61, 233017)-Net in Base 64 — Constructive
(43, 61, 233017)-net in base 64, using
- net defined by OOA [i] based on OOA(6461, 233017, S64, 18, 18), using
- OA 9-folding and stacking [i] based on OA(6461, 2097153, S64, 18), using
- discarding factors based on OA(6461, 2097155, S64, 18), using
- discarding parts of the base [i] based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 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(12849, 2097152, F128, 17) (dual of [2097152, 2097103, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- discarding parts of the base [i] based on linear OA(12852, 2097155, F128, 18) (dual of [2097155, 2097103, 19]-code), using
- discarding factors based on OA(6461, 2097155, S64, 18), using
- OA 9-folding and stacking [i] based on OA(6461, 2097153, S64, 18), using
(61−18, 61, 344837)-Net over F64 — Digital
Digital (43, 61, 344837)-net over F64, using
(61−18, 61, large)-Net in Base 64 — Upper bound on s
There is no (43, 61, large)-net in base 64, because
- 16 times m-reduction [i] would yield (43, 45, large)-net in base 64, but