Best Known (39, 39+35, s)-Nets in Base 64
(39, 39+35, 513)-Net over F64 — Constructive and digital
Digital (39, 74, 513)-net over F64, using
- t-expansion [i] based on digital (28, 74, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
(39, 39+35, 517)-Net in Base 64 — Constructive
(39, 74, 517)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 24, 258)-net in base 64, using
- base change [i] based on digital (1, 18, 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, 18, 258)-net over F256, using
- (15, 50, 259)-net in base 64, using
- 2 times m-reduction [i] based on (15, 52, 259)-net in base 64, using
- base change [i] based on digital (2, 39, 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, 39, 259)-net over F256, using
- 2 times m-reduction [i] based on (15, 52, 259)-net in base 64, using
- (7, 24, 258)-net in base 64, using
(39, 39+35, 2057)-Net over F64 — Digital
Digital (39, 74, 2057)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6474, 2057, F64, 2, 35) (dual of [(2057, 2), 4040, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6474, 4114, F64, 35) (dual of [4114, 4040, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,14]) [i] based on
- linear OA(6469, 4097, F64, 35) (dual of [4097, 4028, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(6457, 4097, F64, 29) (dual of [4097, 4040, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,17]) ⊂ C([0,14]) [i] based on
- OOA 2-folding [i] based on linear OA(6474, 4114, F64, 35) (dual of [4114, 4040, 36]-code), using
(39, 39+35, 6494614)-Net in Base 64 — Upper bound on s
There is no (39, 74, 6494615)-net in base 64, because
- 1 times m-reduction [i] would yield (39, 73, 6494615)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 709803 632896 157749 346330 451098 029910 615439 591054 382617 941797 184430 387369 713145 723037 994075 088430 384186 638188 724513 149338 203149 412850 > 6473 [i]