Best Known (64, 64+19, s)-Nets in Base 16
(64, 64+19, 14580)-Net over F16 — Constructive and digital
Digital (64, 83, 14580)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-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 (0, 9, 17)-net over F16, using
(64, 64+19, 29127)-Net in Base 16 — Constructive
(64, 83, 29127)-net in base 16, using
- net defined by OOA [i] based on OOA(1683, 29127, S16, 19, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(1683, 262144, S16, 19), using
- discarding factors based on OA(1683, 262147, S16, 19), using
- discarding parts of the base [i] based on linear OA(6455, 262147, F64, 19) (dual of [262147, 262092, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(6455, 262144, F64, 19) (dual of [262144, 262089, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(6452, 262144, F64, 18) (dual of [262144, 262092, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(18) ⊂ Ce(17) [i] based on
- discarding parts of the base [i] based on linear OA(6455, 262147, F64, 19) (dual of [262147, 262092, 20]-code), using
- discarding factors based on OA(1683, 262147, S16, 19), using
- OOA 9-folding and stacking with additional row [i] based on OA(1683, 262144, S16, 19), using
(64, 64+19, 179635)-Net over F16 — Digital
Digital (64, 83, 179635)-net over F16, using
(64, 64+19, large)-Net in Base 16 — Upper bound on s
There is no (64, 83, large)-net in base 16, because
- 17 times m-reduction [i] would yield (64, 66, large)-net in base 16, but