Best Known (35−13, 35, s)-Nets in Base 64
(35−13, 35, 827)-Net over F64 — Constructive and digital
Digital (22, 35, 827)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (4, 10, 145)-net over F64, using
- (u, u+v)-construction [i] based on
- 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 (1, 7, 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, 3, 65)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (12, 25, 682)-net over F64, using
- net defined by OOA [i] based on linear OOA(6425, 682, F64, 13, 13) (dual of [(682, 13), 8841, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6425, 4093, F64, 13) (dual of [4093, 4068, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(6425, 4093, F64, 13) (dual of [4093, 4068, 14]-code), using
- net defined by OOA [i] based on linear OOA(6425, 682, F64, 13, 13) (dual of [(682, 13), 8841, 14]-NRT-code), using
- digital (4, 10, 145)-net over F64, using
(35−13, 35, 10923)-Net in Base 64 — Constructive
(22, 35, 10923)-net in base 64, using
- net defined by OOA [i] based on OOA(6435, 10923, S64, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(6435, 65539, S64, 13), using
- discarding factors based on OA(6435, 65542, S64, 13), using
- discarding parts of the base [i] based on linear OA(25626, 65542, F256, 13) (dual of [65542, 65516, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 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(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([0,6]) ⊂ C([0,5]) [i] based on
- discarding parts of the base [i] based on linear OA(25626, 65542, F256, 13) (dual of [65542, 65516, 14]-code), using
- discarding factors based on OA(6435, 65542, S64, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(6435, 65539, S64, 13), using
(35−13, 35, 15567)-Net over F64 — Digital
Digital (22, 35, 15567)-net over F64, using
(35−13, 35, 16384)-Net in Base 64
(22, 35, 16384)-net in base 64, using
- 1 times m-reduction [i] based on (22, 36, 16384)-net in base 64, using
- base change [i] based on digital (13, 27, 16384)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25627, 16384, F256, 4, 14) (dual of [(16384, 4), 65509, 15]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- OOA 4-folding [i] based on linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25627, 16384, F256, 4, 14) (dual of [(16384, 4), 65509, 15]-NRT-code), using
- base change [i] based on digital (13, 27, 16384)-net over F256, using
(35−13, 35, large)-Net in Base 64 — Upper bound on s
There is no (22, 35, large)-net in base 64, because
- 11 times m-reduction [i] would yield (22, 24, large)-net in base 64, but