Best Known (34−14, 34, s)-Nets in Base 64
(34−14, 34, 650)-Net over F64 — Constructive and digital
Digital (20, 34, 650)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 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 (13, 27, 585)-net over F64, using
- net defined by OOA [i] based on linear OOA(6427, 585, F64, 14, 14) (dual of [(585, 14), 8163, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(6427, 4095, F64, 14) (dual of [4095, 4068, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(6427, 4095, F64, 14) (dual of [4095, 4068, 15]-code), using
- net defined by OOA [i] based on linear OOA(6427, 585, F64, 14, 14) (dual of [(585, 14), 8163, 15]-NRT-code), using
- digital (0, 7, 65)-net over F64, using
(34−14, 34, 2341)-Net in Base 64 — Constructive
(20, 34, 2341)-net in base 64, using
- 1 times m-reduction [i] based on (20, 35, 2341)-net in base 64, using
- base change [i] based on digital (15, 30, 2341)-net over F128, using
- net defined by OOA [i] based on linear OOA(12830, 2341, F128, 15, 15) (dual of [(2341, 15), 35085, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12830, 16388, F128, 15) (dual of [16388, 16358, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12830, 16388, F128, 15) (dual of [16388, 16358, 16]-code), using
- net defined by OOA [i] based on linear OOA(12830, 2341, F128, 15, 15) (dual of [(2341, 15), 35085, 16]-NRT-code), using
- base change [i] based on digital (15, 30, 2341)-net over F128, using
(34−14, 34, 5061)-Net over F64 — Digital
Digital (20, 34, 5061)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6434, 5061, F64, 14) (dual of [5061, 5027, 15]-code), using
- 956 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 12 times 0, 1, 53 times 0, 1, 207 times 0, 1, 677 times 0) [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- 956 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 12 times 0, 1, 53 times 0, 1, 207 times 0, 1, 677 times 0) [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
(34−14, 34, 5463)-Net in Base 64
(20, 34, 5463)-net in base 64, using
- 1 times m-reduction [i] based on (20, 35, 5463)-net in base 64, using
- base change [i] based on digital (15, 30, 5463)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12830, 5463, F128, 3, 15) (dual of [(5463, 3), 16359, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12830, 16389, F128, 15) (dual of [16389, 16359, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12830, 16390, F128, 15) (dual of [16390, 16360, 16]-code), using
- OOA 3-folding [i] based on linear OA(12830, 16389, F128, 15) (dual of [16389, 16359, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12830, 5463, F128, 3, 15) (dual of [(5463, 3), 16359, 16]-NRT-code), using
- base change [i] based on digital (15, 30, 5463)-net over F128, using
(34−14, 34, large)-Net in Base 64 — Upper bound on s
There is no (20, 34, large)-net in base 64, because
- 12 times m-reduction [i] would yield (20, 22, large)-net in base 64, but