Best Known (30−11, 30, s)-Nets in Base 64
(30−11, 30, 4175)-Net over F64 — Constructive and digital
Digital (19, 30, 4175)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 65)-net over F64, using
- s-reduction based on digital (0, 0, s)-net over F64 with arbitrarily large s, using
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- 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, 1, 65)-net over F64 (see above)
- 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, 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, 5, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (1, 12, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- digital (0, 0, 65)-net over F64, using
(30−11, 30, 13108)-Net in Base 64 — Constructive
(19, 30, 13108)-net in base 64, using
- net defined by OOA [i] based on OOA(6430, 13108, S64, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(6430, 65541, S64, 11), using
- discarding factors based on OA(6430, 65542, S64, 11), using
- discarding parts of the base [i] based on linear OA(25622, 65542, F256, 11) (dual of [65542, 65520, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(25621, 65537, F256, 11) (dual of [65537, 65516, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- discarding parts of the base [i] based on linear OA(25622, 65542, F256, 11) (dual of [65542, 65520, 12]-code), using
- discarding factors based on OA(6430, 65542, S64, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(6430, 65541, S64, 11), using
(30−11, 30, 18849)-Net over F64 — Digital
Digital (19, 30, 18849)-net over F64, using
(30−11, 30, 20651)-Net in Base 64
(19, 30, 20651)-net in base 64, using
- 642 times duplication [i] based on (17, 28, 20651)-net in base 64, using
- base change [i] based on digital (10, 21, 20651)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25621, 20651, F256, 3, 11) (dual of [(20651, 3), 61932, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25621, 21846, F256, 3, 11) (dual of [(21846, 3), 65517, 12]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25621, 65538, F256, 11) (dual of [65538, 65517, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(25621, 65536, F256, 11) (dual of [65536, 65515, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- OOA 3-folding [i] based on linear OA(25621, 65538, F256, 11) (dual of [65538, 65517, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(25621, 21846, F256, 3, 11) (dual of [(21846, 3), 65517, 12]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25621, 20651, F256, 3, 11) (dual of [(20651, 3), 61932, 12]-NRT-code), using
- base change [i] based on digital (10, 21, 20651)-net over F256, using
(30−11, 30, large)-Net in Base 64 — Upper bound on s
There is no (19, 30, large)-net in base 64, because
- 9 times m-reduction [i] would yield (19, 21, large)-net in base 64, but