Best Known (55, 77, s)-Nets in Base 64
(55, 77, 23911)-Net over F64 — Constructive and digital
Digital (55, 77, 23911)-net over F64, using
- 641 times duplication [i] based on digital (54, 76, 23911)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 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 (42, 64, 23831)-net over F64, using
- net defined by OOA [i] based on linear OOA(6464, 23831, F64, 22, 22) (dual of [(23831, 22), 524218, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(6464, 262141, F64, 22) (dual of [262141, 262077, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(6464, 262144, F64, 22) (dual of [262144, 262080, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(6464, 262141, F64, 22) (dual of [262141, 262077, 23]-code), using
- net defined by OOA [i] based on linear OOA(6464, 23831, F64, 22, 22) (dual of [(23831, 22), 524218, 23]-NRT-code), using
- digital (1, 12, 80)-net over F64, using
- (u, u+v)-construction [i] based on
(55, 77, 190651)-Net in Base 64 — Constructive
(55, 77, 190651)-net in base 64, using
- base change [i] based on digital (44, 66, 190651)-net over F128, using
- net defined by OOA [i] based on linear OOA(12866, 190651, F128, 22, 22) (dual of [(190651, 22), 4194256, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(12866, 2097161, F128, 22) (dual of [2097161, 2097095, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(12866, 2097163, F128, 22) (dual of [2097163, 2097097, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(12864, 2097152, F128, 22) (dual of [2097152, 2097088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(12866, 2097163, F128, 22) (dual of [2097163, 2097097, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(12866, 2097161, F128, 22) (dual of [2097161, 2097095, 23]-code), using
- net defined by OOA [i] based on linear OOA(12866, 190651, F128, 22, 22) (dual of [(190651, 22), 4194256, 23]-NRT-code), using
(55, 77, 577856)-Net over F64 — Digital
Digital (55, 77, 577856)-net over F64, using
(55, 77, 782307)-Net in Base 64
(55, 77, 782307)-net in base 64, using
- base change [i] based on digital (44, 66, 782307)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12866, 782307, F128, 2, 22) (dual of [(782307, 2), 1564548, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(12866, 1048581, F128, 2, 22) (dual of [(1048581, 2), 2097096, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(12866, 2097162, F128, 22) (dual of [2097162, 2097096, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(12866, 2097163, F128, 22) (dual of [2097163, 2097097, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(12864, 2097152, F128, 22) (dual of [2097152, 2097088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(12855, 2097152, F128, 19) (dual of [2097152, 2097097, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(12866, 2097163, F128, 22) (dual of [2097163, 2097097, 23]-code), using
- OOA 2-folding [i] based on linear OA(12866, 2097162, F128, 22) (dual of [2097162, 2097096, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(12866, 1048581, F128, 2, 22) (dual of [(1048581, 2), 2097096, 23]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12866, 782307, F128, 2, 22) (dual of [(782307, 2), 1564548, 23]-NRT-code), using
(55, 77, large)-Net in Base 64 — Upper bound on s
There is no (55, 77, large)-net in base 64, because
- 20 times m-reduction [i] would yield (55, 57, large)-net in base 64, but