Best Known (27, 27+23, s)-Nets in Base 64
(27, 27+23, 373)-Net over F64 — Constructive and digital
Digital (27, 50, 373)-net over F64, using
- 643 times duplication [i] based on digital (24, 47, 373)-net over F64, using
- net defined by OOA [i] based on linear OOA(6447, 373, F64, 23, 23) (dual of [(373, 23), 8532, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(6447, 4104, F64, 23) (dual of [4104, 4057, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(6447, 4104, F64, 23) (dual of [4104, 4057, 24]-code), using
- net defined by OOA [i] based on linear OOA(6447, 373, F64, 23, 23) (dual of [(373, 23), 8532, 24]-NRT-code), using
(27, 27+23, 517)-Net in Base 64 — Constructive
(27, 50, 517)-net in base 64, using
- (u, u+v)-construction [i] based on
- (5, 16, 258)-net in base 64, using
- base change [i] based on digital (1, 12, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- base change [i] based on digital (1, 12, 258)-net over F256, using
- (11, 34, 259)-net in base 64, using
- 2 times m-reduction [i] based on (11, 36, 259)-net in base 64, using
- base change [i] based on digital (2, 27, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- base change [i] based on digital (2, 27, 259)-net over F256, using
- 2 times m-reduction [i] based on (11, 36, 259)-net in base 64, using
- (5, 16, 258)-net in base 64, using
(27, 27+23, 2248)-Net over F64 — Digital
Digital (27, 50, 2248)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6450, 2248, F64, 23) (dual of [2248, 2198, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(6450, 4114, F64, 23) (dual of [4114, 4064, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(6445, 4097, F64, 23) (dual of [4097, 4052, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(645, 17, F64, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,64)), using
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- Reed–Solomon code RS(59,64) [i]
- discarding factors / shortening the dual code based on linear OA(645, 64, F64, 5) (dual of [64, 59, 6]-code or 64-arc in PG(4,64)), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6450, 4114, F64, 23) (dual of [4114, 4064, 24]-code), using
(27, 27+23, large)-Net in Base 64 — Upper bound on s
There is no (27, 50, large)-net in base 64, because
- 21 times m-reduction [i] would yield (27, 29, large)-net in base 64, but