Best Known (66, 66+19, s)-Nets in Base 16
(66, 66+19, 14596)-Net over F16 — Constructive and digital
Digital (66, 85, 14596)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 11, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- digital (55, 74, 14563)-net over F16, using
- net defined by OOA [i] based on linear OOA(1674, 14563, F16, 19, 19) (dual of [(14563, 19), 276623, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1674, 131068, F16, 19) (dual of [131068, 130994, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1674, 131074, F16, 19) (dual of [131074, 131000, 20]-code), using
- trace code [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]
- trace code [i] based on linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(1674, 131074, F16, 19) (dual of [131074, 131000, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(1674, 131068, F16, 19) (dual of [131068, 130994, 20]-code), using
- net defined by OOA [i] based on linear OOA(1674, 14563, F16, 19, 19) (dual of [(14563, 19), 276623, 20]-NRT-code), using
- digital (2, 11, 33)-net over F16, using
(66, 66+19, 29128)-Net in Base 16 — Constructive
(66, 85, 29128)-net in base 16, using
- net defined by OOA [i] based on OOA(1685, 29128, S16, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(1685, 262153, S16, 19), using
- 1 times code embedding in larger space [i] based on OA(1684, 262152, S16, 19), using
- discarding parts of the base [i] based on linear OA(6456, 262152, F64, 19) (dual of [262152, 262096, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(6455, 262145, F64, 19) (dual of [262145, 262090, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(6449, 262145, F64, 17) (dual of [262145, 262096, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 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 parts of the base [i] based on linear OA(6456, 262152, F64, 19) (dual of [262152, 262096, 20]-code), using
- 1 times code embedding in larger space [i] based on OA(1684, 262152, S16, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(1685, 262153, S16, 19), using
(66, 66+19, 244443)-Net over F16 — Digital
Digital (66, 85, 244443)-net over F16, using
(66, 66+19, large)-Net in Base 16 — Upper bound on s
There is no (66, 85, large)-net in base 16, because
- 17 times m-reduction [i] would yield (66, 68, large)-net in base 16, but