Best Known (58, 58+15, s)-Nets in Base 64
(58, 58+15, 1202531)-Net over F64 — Constructive and digital
Digital (58, 73, 1202531)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (9, 16, 4160)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 65)-net over F64, using
- s-reduction based on digital (0, 0, s)-net over F64 with arbitrarily large s, using
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 1, 65)-net over F64, using
- s-reduction based on digital (0, 1, s)-net over F64 with arbitrarily large s, using
- digital (0, 1, 65)-net over F64 (see above)
- digital (0, 1, 65)-net over F64 (see above)
- digital (0, 1, 65)-net over F64 (see above)
- digital (0, 2, 65)-net over F64, using
- digital (0, 3, 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 (0, 7, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 0, 65)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (42, 57, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- digital (9, 16, 4160)-net over F64, using
(58, 58+15, 1203832)-Net in Base 64 — Constructive
(58, 73, 1203832)-net in base 64, using
- (u, u+v)-construction [i] based on
- (9, 16, 5461)-net in base 64, using
- net defined by OOA [i] based on OOA(6416, 5461, S64, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(6416, 16384, S64, 7), using
- discarding factors based on OA(6416, 16386, S64, 7), using
- discarding parts of the base [i] based on linear OA(12813, 16386, F128, 7) (dual of [16386, 16373, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(12811, 16384, F128, 6) (dual of [16384, 16373, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 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(6) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(12813, 16386, F128, 7) (dual of [16386, 16373, 8]-code), using
- discarding factors based on OA(6416, 16386, S64, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(6416, 16384, S64, 7), using
- net defined by OOA [i] based on OOA(6416, 5461, S64, 7, 7), using
- digital (42, 57, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- (9, 16, 5461)-net in base 64, using
(58, 58+15, large)-Net over F64 — Digital
Digital (58, 73, large)-net over F64, using
- t-expansion [i] based on digital (57, 73, large)-net over F64, using
- 3 times m-reduction [i] based on digital (57, 76, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6476, large, F64, 19) (dual of [large, large−76, 20]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 3 times code embedding in larger space [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6476, large, F64, 19) (dual of [large, large−76, 20]-code), using
- 3 times m-reduction [i] based on digital (57, 76, large)-net over F64, using
(58, 58+15, large)-Net in Base 64 — Upper bound on s
There is no (58, 73, large)-net in base 64, because
- 13 times m-reduction [i] would yield (58, 60, large)-net in base 64, but