Best Known (52, 52+19, s)-Nets in Base 64
(52, 52+19, 29322)-Net over F64 — Constructive and digital
Digital (52, 71, 29322)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (7, 16, 195)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- 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, 4, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 9, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 3, 65)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (36, 55, 29127)-net over F64, using
- net defined by OOA [i] based on linear OOA(6455, 29127, F64, 19, 19) (dual of [(29127, 19), 553358, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(6455, 262144, F64, 19) (dual of [262144, 262089, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(6455, 262144, F64, 19) (dual of [262144, 262089, 20]-code), using
- net defined by OOA [i] based on linear OOA(6455, 29127, F64, 19, 19) (dual of [(29127, 19), 553358, 20]-NRT-code), using
- digital (7, 16, 195)-net over F64, using
(52, 52+19, 233019)-Net in Base 64 — Constructive
(52, 71, 233019)-net in base 64, using
- 641 times duplication [i] based on (51, 70, 233019)-net in base 64, using
- base change [i] based on digital (41, 60, 233019)-net over F128, using
- net defined by OOA [i] based on linear OOA(12860, 233019, F128, 19, 19) (dual of [(233019, 19), 4427301, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12860, 2097172, F128, 19) (dual of [2097172, 2097112, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(12860, 2097176, F128, 19) (dual of [2097176, 2097116, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(12855, 2097153, F128, 19) (dual of [2097153, 2097098, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(12837, 2097153, F128, 13) (dual of [2097153, 2097116, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1285, 23, F128, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12860, 2097176, F128, 19) (dual of [2097176, 2097116, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(12860, 2097172, F128, 19) (dual of [2097172, 2097112, 20]-code), using
- net defined by OOA [i] based on linear OOA(12860, 233019, F128, 19, 19) (dual of [(233019, 19), 4427301, 20]-NRT-code), using
- base change [i] based on digital (41, 60, 233019)-net over F128, using
(52, 52+19, 1596499)-Net over F64 — Digital
Digital (52, 71, 1596499)-net over F64, using
(52, 52+19, large)-Net in Base 64 — Upper bound on s
There is no (52, 71, large)-net in base 64, because
- 17 times m-reduction [i] would yield (52, 54, large)-net in base 64, but