Best Known (28, 28+17, s)-Nets in Base 64
(28, 28+17, 650)-Net over F64 — Constructive and digital
Digital (28, 45, 650)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 65)-net over F64, using
- s-reduction based on digital (0, 1, s)-net over F64 with arbitrarily large s, using
- digital (0, 1, 65)-net over F64 (see above)
- digital (0, 2, 65)-net over F64, using
- digital (0, 2, 65)-net over F64 (see above)
- digital (0, 2, 65)-net over F64 (see above)
- digital (0, 3, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (0, 4, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 5, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 8, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 17, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 1, 65)-net over F64, using
(28, 28+17, 8192)-Net in Base 64 — Constructive
(28, 45, 8192)-net in base 64, using
- 641 times duplication [i] based on (27, 44, 8192)-net in base 64, using
- base change [i] based on digital (16, 33, 8192)-net over F256, using
- net defined by OOA [i] based on linear OOA(25633, 8192, F256, 17, 17) (dual of [(8192, 17), 139231, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(25633, 65537, F256, 17) (dual of [65537, 65504, 18]-code), using
- net defined by OOA [i] based on linear OOA(25633, 8192, F256, 17, 17) (dual of [(8192, 17), 139231, 18]-NRT-code), using
- base change [i] based on digital (16, 33, 8192)-net over F256, using
(28, 28+17, 12982)-Net over F64 — Digital
Digital (28, 45, 12982)-net over F64, using
(28, 28+17, 13695)-Net in Base 64
(28, 45, 13695)-net in base 64, using
- 641 times duplication [i] based on (27, 44, 13695)-net in base 64, using
- base change [i] based on digital (16, 33, 13695)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25633, 13695, F256, 4, 17) (dual of [(13695, 4), 54747, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25633, 16384, F256, 4, 17) (dual of [(16384, 4), 65503, 18]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25633, 65536, F256, 17) (dual of [65536, 65503, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 4-folding [i] based on linear OA(25633, 65536, F256, 17) (dual of [65536, 65503, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(25633, 16384, F256, 4, 17) (dual of [(16384, 4), 65503, 18]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25633, 13695, F256, 4, 17) (dual of [(13695, 4), 54747, 18]-NRT-code), using
- base change [i] based on digital (16, 33, 13695)-net over F256, using
(28, 28+17, large)-Net in Base 64 — Upper bound on s
There is no (28, 45, large)-net in base 64, because
- 15 times m-reduction [i] would yield (28, 30, large)-net in base 64, but