Best Known (65−18, 65, s)-Nets in Base 64
(65−18, 65, 29257)-Net over F64 — Constructive and digital
Digital (47, 65, 29257)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 130)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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, 9, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 4, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- 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 (4, 13, 130)-net over F64, using
(65−18, 65, 233018)-Net in Base 64 — Constructive
(47, 65, 233018)-net in base 64, using
- 642 times duplication [i] based on (45, 63, 233018)-net in base 64, using
- base change [i] based on digital (36, 54, 233018)-net over F128, using
- net defined by OOA [i] based on linear OOA(12854, 233018, F128, 18, 18) (dual of [(233018, 18), 4194270, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(12854, 2097162, F128, 18) (dual of [2097162, 2097108, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12854, 2097163, F128, 18) (dual of [2097163, 2097109, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [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(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(12854, 2097163, F128, 18) (dual of [2097163, 2097109, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(12854, 2097162, F128, 18) (dual of [2097162, 2097108, 19]-code), using
- net defined by OOA [i] based on linear OOA(12854, 233018, F128, 18, 18) (dual of [(233018, 18), 4194270, 19]-NRT-code), using
- base change [i] based on digital (36, 54, 233018)-net over F128, using
(65−18, 65, 917468)-Net over F64 — Digital
Digital (47, 65, 917468)-net over F64, using
(65−18, 65, 1020833)-Net in Base 64
(47, 65, 1020833)-net in base 64, using
- 642 times duplication [i] based on (45, 63, 1020833)-net in base 64, using
- base change [i] based on digital (36, 54, 1020833)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12854, 1020833, F128, 2, 18) (dual of [(1020833, 2), 2041612, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12854, 1048581, F128, 2, 18) (dual of [(1048581, 2), 2097108, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12854, 2097162, F128, 18) (dual of [2097162, 2097108, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(12854, 2097163, F128, 18) (dual of [2097163, 2097109, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [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(12843, 2097152, F128, 15) (dual of [2097152, 2097109, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(12854, 2097163, F128, 18) (dual of [2097163, 2097109, 19]-code), using
- OOA 2-folding [i] based on linear OA(12854, 2097162, F128, 18) (dual of [2097162, 2097108, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(12854, 1048581, F128, 2, 18) (dual of [(1048581, 2), 2097108, 19]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12854, 1020833, F128, 2, 18) (dual of [(1020833, 2), 2041612, 19]-NRT-code), using
- base change [i] based on digital (36, 54, 1020833)-net over F128, using
(65−18, 65, large)-Net in Base 64 — Upper bound on s
There is no (47, 65, large)-net in base 64, because
- 16 times m-reduction [i] would yield (47, 49, large)-net in base 64, but