Best Known (71, 71+21, s)-Nets in Base 16
(71, 71+21, 13124)-Net over F16 — Constructive and digital
Digital (71, 92, 13124)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 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 (61, 82, 13107)-net over F16, using
- net defined by OOA [i] based on linear OOA(1682, 13107, F16, 21, 21) (dual of [(13107, 21), 275165, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1682, 131071, F16, 21) (dual of [131071, 130989, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1682, 131074, F16, 21) (dual of [131074, 130992, 22]-code), using
- trace code [i] based on linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- trace code [i] based on linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1682, 131074, F16, 21) (dual of [131074, 130992, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1682, 131071, F16, 21) (dual of [131071, 130989, 22]-code), using
- net defined by OOA [i] based on linear OOA(1682, 13107, F16, 21, 21) (dual of [(13107, 21), 275165, 22]-NRT-code), using
- digital (0, 10, 17)-net over F16, using
(71, 71+21, 26214)-Net in Base 16 — Constructive
(71, 92, 26214)-net in base 16, using
- net defined by OOA [i] based on OOA(1692, 26214, S16, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(1692, 262141, S16, 21), using
- discarding factors based on OA(1692, 262147, S16, 21), using
- discarding parts of the base [i] based on linear OA(6461, 262147, F64, 21) (dual of [262147, 262086, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(20) ⊂ Ce(19) [i] based on
- discarding parts of the base [i] based on linear OA(6461, 262147, F64, 21) (dual of [262147, 262086, 22]-code), using
- discarding factors based on OA(1692, 262147, S16, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(1692, 262141, S16, 21), using
(71, 71+21, 191509)-Net over F16 — Digital
Digital (71, 92, 191509)-net over F16, using
(71, 71+21, large)-Net in Base 16 — Upper bound on s
There is no (71, 92, large)-net in base 16, because
- 19 times m-reduction [i] would yield (71, 73, large)-net in base 16, but