Best Known (21, 38, s)-Nets in Base 64
(21, 38, 514)-Net over F64 — Constructive and digital
Digital (21, 38, 514)-net over F64, using
- net defined by OOA [i] based on linear OOA(6438, 514, F64, 17, 17) (dual of [(514, 17), 8700, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6438, 4113, F64, 17) (dual of [4113, 4075, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6438, 4114, F64, 17) (dual of [4114, 4076, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- 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(6421, 4097, F64, 11) (dual of [4097, 4076, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6438, 4114, F64, 17) (dual of [4114, 4076, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6438, 4113, F64, 17) (dual of [4113, 4075, 18]-code), using
(21, 38, 517)-Net in Base 64 — Constructive
(21, 38, 517)-net in base 64, using
- (u, u+v)-construction [i] based on
- (4, 12, 258)-net in base 64, using
- base change [i] based on digital (1, 9, 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, 9, 258)-net over F256, using
- (9, 26, 259)-net in base 64, using
- 2 times m-reduction [i] based on (9, 28, 259)-net in base 64, using
- base change [i] based on digital (2, 21, 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, 21, 259)-net over F256, using
- 2 times m-reduction [i] based on (9, 28, 259)-net in base 64, using
- (4, 12, 258)-net in base 64, using
(21, 38, 2902)-Net over F64 — Digital
Digital (21, 38, 2902)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6438, 2902, F64, 17) (dual of [2902, 2864, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6438, 4114, F64, 17) (dual of [4114, 4076, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- 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(6421, 4097, F64, 11) (dual of [4097, 4076, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6438, 4114, F64, 17) (dual of [4114, 4076, 18]-code), using
(21, 38, large)-Net in Base 64 — Upper bound on s
There is no (21, 38, large)-net in base 64, because
- 15 times m-reduction [i] would yield (21, 23, large)-net in base 64, but