Best Known (20, 20+8, s)-Nets in Base 64
(20, 20+8, 67552)-Net over F64 — Constructive and digital
Digital (20, 28, 67552)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 2016)-net over F64, using
- net defined by OOA [i] based on linear OOA(646, 2016, F64, 4, 4) (dual of [(2016, 4), 8058, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(646, 2016, F64, 3, 4) (dual of [(2016, 3), 6042, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(646, 4032, F64, 4) (dual of [4032, 4026, 5]-code), using
- 1 times truncation [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(646, 4032, F64, 4) (dual of [4032, 4026, 5]-code), using
- appending kth column [i] based on linear OOA(646, 2016, F64, 3, 4) (dual of [(2016, 3), 6042, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(646, 2016, F64, 4, 4) (dual of [(2016, 4), 8058, 5]-NRT-code), using
- digital (14, 22, 65536)-net over F64, using
- net defined by OOA [i] based on linear OOA(6422, 65536, F64, 8, 8) (dual of [(65536, 8), 524266, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OA 4-folding and stacking [i] based on linear OA(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using
- net defined by OOA [i] based on linear OOA(6422, 65536, F64, 8, 8) (dual of [(65536, 8), 524266, 9]-NRT-code), using
- digital (2, 6, 2016)-net over F64, using
(20, 20+8, 524290)-Net in Base 64 — Constructive
(20, 28, 524290)-net in base 64, using
- base change [i] based on digital (16, 24, 524290)-net over F128, using
- net defined by OOA [i] based on linear OOA(12824, 524290, F128, 8, 8) (dual of [(524290, 8), 4194296, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(12824, 2097160, F128, 8) (dual of [2097160, 2097136, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(12824, 2097163, F128, 8) (dual of [2097163, 2097139, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(12824, 2097163, F128, 8) (dual of [2097163, 2097139, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(12824, 2097160, F128, 8) (dual of [2097160, 2097136, 9]-code), using
- net defined by OOA [i] based on linear OOA(12824, 524290, F128, 8, 8) (dual of [(524290, 8), 4194296, 9]-NRT-code), using
(20, 20+8, 900118)-Net over F64 — Digital
Digital (20, 28, 900118)-net over F64, using
(20, 20+8, 2097163)-Net in Base 64
(20, 28, 2097163)-net in base 64, using
- base change [i] based on digital (16, 24, 2097163)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12824, 2097163, F128, 8) (dual of [2097163, 2097139, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(12822, 2097152, F128, 8) (dual of [2097152, 2097130, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(7) ⊂ Ce(4) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12824, 2097163, F128, 8) (dual of [2097163, 2097139, 9]-code), using
(20, 20+8, large)-Net in Base 64 — Upper bound on s
There is no (20, 28, large)-net in base 64, because
- 6 times m-reduction [i] would yield (20, 22, large)-net in base 64, but