Best Known (45, 45+45, s)-Nets in Base 64
(45, 45+45, 513)-Net over F64 — Constructive and digital
Digital (45, 90, 513)-net over F64, using
- t-expansion [i] based on digital (28, 90, 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
(45, 45+45, 514)-Net in Base 64 — Constructive
(45, 90, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (8, 30, 257)-net in base 64, using
- 2 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- base change [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- 2 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
- (15, 60, 257)-net in base 64, using
- base change [i] based on digital (0, 45, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- base change [i] based on digital (0, 45, 257)-net over F256, using
- (8, 30, 257)-net in base 64, using
(45, 45+45, 1575)-Net over F64 — Digital
Digital (45, 90, 1575)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6490, 1575, F64, 2, 45) (dual of [(1575, 2), 3060, 46]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6490, 2051, F64, 2, 45) (dual of [(2051, 2), 4012, 46]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6490, 4102, F64, 45) (dual of [4102, 4012, 46]-code), using
- construction X applied to C([0,22]) ⊂ C([0,21]) [i] based on
- linear OA(6489, 4097, F64, 45) (dual of [4097, 4008, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- linear OA(6485, 4097, F64, 43) (dual of [4097, 4012, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,22]) ⊂ C([0,21]) [i] based on
- OOA 2-folding [i] based on linear OA(6490, 4102, F64, 45) (dual of [4102, 4012, 46]-code), using
- discarding factors / shortening the dual code based on linear OOA(6490, 2051, F64, 2, 45) (dual of [(2051, 2), 4012, 46]-NRT-code), using
(45, 45+45, 2912928)-Net in Base 64 — Upper bound on s
There is no (45, 90, 2912929)-net in base 64, because
- 1 times m-reduction [i] would yield (45, 89, 2912929)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 56236 575158 413650 904890 376484 452088 738699 830405 103231 360492 907082 130946 278936 208677 561211 019349 223997 027545 931538 679043 047792 705262 008176 796696 330760 363064 207920 > 6489 [i]