Best Known (44, 44+43, s)-Nets in Base 64
(44, 44+43, 513)-Net over F64 — Constructive and digital
Digital (44, 87, 513)-net over F64, using
- t-expansion [i] based on digital (28, 87, 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
(44, 44+43, 515)-Net in Base 64 — Constructive
(44, 87, 515)-net in base 64, using
- (u, u+v)-construction [i] based on
- (7, 28, 257)-net in base 64, using
- base change [i] based on digital (0, 21, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 21, 257)-net over F256, using
- (16, 59, 258)-net in base 64, using
- 1 times m-reduction [i] based on (16, 60, 258)-net in base 64, using
- base change [i] based on digital (1, 45, 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, 45, 258)-net over F256, using
- 1 times m-reduction [i] based on (16, 60, 258)-net in base 64, using
- (7, 28, 257)-net in base 64, using
(44, 44+43, 1705)-Net over F64 — Digital
Digital (44, 87, 1705)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6487, 1705, F64, 2, 43) (dual of [(1705, 2), 3323, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6487, 2052, F64, 2, 43) (dual of [(2052, 2), 4017, 44]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6487, 4104, F64, 43) (dual of [4104, 4017, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(39) [i] based on
- linear OA(6485, 4096, F64, 43) (dual of [4096, 4011, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(6479, 4096, F64, 40) (dual of [4096, 4017, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [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(42) ⊂ Ce(39) [i] based on
- OOA 2-folding [i] based on linear OA(6487, 4104, F64, 43) (dual of [4104, 4017, 44]-code), using
- discarding factors / shortening the dual code based on linear OOA(6487, 2052, F64, 2, 43) (dual of [(2052, 2), 4017, 44]-NRT-code), using
(44, 44+43, 3434691)-Net in Base 64 — Upper bound on s
There is no (44, 87, 3434692)-net in base 64, because
- 1 times m-reduction [i] would yield (44, 86, 3434692)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 214525 456352 175189 168272 584370 559532 952475 496837 945370 684926 242888 461087 922265 908047 072481 980963 627623 043262 254174 221157 067163 152566 559345 045644 151233 792368 > 6486 [i]