Best Known (45, 45+18, s)-Nets in Base 64
(45, 45+18, 29207)-Net over F64 — Constructive and digital
Digital (45, 63, 29207)-net over F64, using
- 641 times duplication [i] based on digital (44, 62, 29207)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-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 (1, 10, 80)-net over F64, using
- (u, u+v)-construction [i] based on
(45, 45+18, 233018)-Net in Base 64 — Constructive
(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
(45, 45+18, 562473)-Net over F64 — Digital
Digital (45, 63, 562473)-net over F64, using
(45, 45+18, 1020833)-Net in Base 64
(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
(45, 45+18, large)-Net in Base 64 — Upper bound on s
There is no (45, 63, large)-net in base 64, because
- 16 times m-reduction [i] would yield (45, 47, large)-net in base 64, but