Best Known (19−6, 19, s)-Nets in Base 64
(19−6, 19, 87447)-Net over F64 — Constructive and digital
Digital (13, 19, 87447)-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 (10, 16, 87382)-net over F64, using
- net defined by OOA [i] based on linear OOA(6416, 87382, F64, 6, 6) (dual of [(87382, 6), 524276, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(6416, 262146, F64, 6) (dual of [262146, 262130, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(6416, 262147, F64, 6) (dual of [262147, 262131, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(6413, 262144, F64, 5) (dual of [262144, 262131, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(6416, 262147, F64, 6) (dual of [262147, 262131, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(6416, 262146, F64, 6) (dual of [262146, 262130, 7]-code), using
- net defined by OOA [i] based on linear OOA(6416, 87382, F64, 6, 6) (dual of [(87382, 6), 524276, 7]-NRT-code), using
- digital (0, 3, 65)-net over F64, using
(19−6, 19, 301983)-Net over F64 — Digital
Digital (13, 19, 301983)-net over F64, using
(19−6, 19, 699051)-Net in Base 64 — Constructive
(13, 19, 699051)-net in base 64, using
- net defined by OOA [i] based on OOA(6419, 699051, S64, 6, 6), using
- OA 3-folding and stacking [i] based on OA(6419, 2097153, S64, 6), using
- discarding factors based on OA(6419, 2097155, S64, 6), using
- discarding parts of the base [i] based on linear OA(12816, 2097155, F128, 6) (dual of [2097155, 2097139, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(12813, 2097152, F128, 5) (dual of [2097152, 2097139, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding parts of the base [i] based on linear OA(12816, 2097155, F128, 6) (dual of [2097155, 2097139, 7]-code), using
- discarding factors based on OA(6419, 2097155, S64, 6), using
- OA 3-folding and stacking [i] based on OA(6419, 2097153, S64, 6), using
(19−6, 19, 1390868)-Net in Base 64
(13, 19, 1390868)-net in base 64, using
- net defined by OOA [i] based on OOA(6419, 1390868, S64, 9, 6), using
- OOA stacking with additional row [i] based on OOA(6419, 1390869, S64, 3, 6), using
- discarding parts of the base [i] based on linear OOA(12816, 1390869, F128, 3, 6) (dual of [(1390869, 3), 4172591, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12816, 1390869, F128, 6) (dual of [1390869, 1390853, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12816, 1390869, F128, 6) (dual of [1390869, 1390853, 7]-code), using
- discarding parts of the base [i] based on linear OOA(12816, 1390869, F128, 3, 6) (dual of [(1390869, 3), 4172591, 7]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(6419, 1390869, S64, 3, 6), using
(19−6, 19, large)-Net in Base 64 — Upper bound on s
There is no (13, 19, large)-net in base 64, because
- 4 times m-reduction [i] would yield (13, 15, large)-net in base 64, but