Best Known (53−19, 53, s)-Nets in Base 64
(53−19, 53, 780)-Net over F64 — Constructive and digital
Digital (34, 53, 780)-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, 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, 3, 65)-net over F64 (see above)
- digital (0, 4, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 6, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 9, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 19, 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
(53−19, 53, 7282)-Net in Base 64 — Constructive
(34, 53, 7282)-net in base 64, using
- 641 times duplication [i] based on (33, 52, 7282)-net in base 64, using
- base change [i] based on digital (20, 39, 7282)-net over F256, using
- 2561 times duplication [i] based on digital (19, 38, 7282)-net over F256, using
- net defined by OOA [i] based on linear OOA(25638, 7282, F256, 19, 19) (dual of [(7282, 19), 138320, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25638, 65539, F256, 19) (dual of [65539, 65501, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(25638, 65542, F256, 19) (dual of [65542, 65504, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 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]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25638, 65542, F256, 19) (dual of [65542, 65504, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(25638, 65539, F256, 19) (dual of [65539, 65501, 20]-code), using
- net defined by OOA [i] based on linear OOA(25638, 7282, F256, 19, 19) (dual of [(7282, 19), 138320, 20]-NRT-code), using
- 2561 times duplication [i] based on digital (19, 38, 7282)-net over F256, using
- base change [i] based on digital (20, 39, 7282)-net over F256, using
(53−19, 53, 24954)-Net over F64 — Digital
Digital (34, 53, 24954)-net over F64, using
(53−19, 53, large)-Net in Base 64 — Upper bound on s
There is no (34, 53, large)-net in base 64, because
- 17 times m-reduction [i] would yield (34, 36, large)-net in base 64, but