Best Known (33, 33+33, s)-Nets in Base 64
(33, 33+33, 513)-Net over F64 — Constructive and digital
Digital (33, 66, 513)-net over F64, using
- t-expansion [i] based on digital (28, 66, 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
(33, 33+33, 514)-Net in Base 64 — Constructive
(33, 66, 514)-net in base 64, using
- (u, u+v)-construction [i] based on
- (6, 22, 257)-net in base 64, using
- 2 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- base change [i] based on digital (0, 18, 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, 18, 257)-net over F256, using
- 2 times m-reduction [i] based on (6, 24, 257)-net in base 64, using
- (11, 44, 257)-net in base 64, using
- base change [i] based on digital (0, 33, 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, 33, 257)-net over F256, using
- (6, 22, 257)-net in base 64, using
(33, 33+33, 1367)-Net over F64 — Digital
Digital (33, 66, 1367)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(6466, 1367, F64, 3, 33) (dual of [(1367, 3), 4035, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6466, 4101, F64, 33) (dual of [4101, 4035, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(6466, 4102, F64, 33) (dual of [4102, 4036, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- linear OA(6465, 4097, F64, 33) (dual of [4097, 4032, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(6461, 4097, F64, 31) (dual of [4097, 4036, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (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([0,16]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6466, 4102, F64, 33) (dual of [4102, 4036, 34]-code), using
- OOA 3-folding [i] based on linear OA(6466, 4101, F64, 33) (dual of [4101, 4035, 34]-code), using
(33, 33+33, 2348566)-Net in Base 64 — Upper bound on s
There is no (33, 66, 2348567)-net in base 64, because
- 1 times m-reduction [i] would yield (33, 65, 2348567)-net in base 64, but
- the generalized Rao bound for nets shows that 64m ≥ 2521 734774 321405 441396 428338 367761 409149 567523 235722 363996 203829 177781 544031 444928 388221 813313 566331 287609 283803 079189 > 6465 [i]