Best Known (15, 15+19, s)-Nets in Base 16
(15, 15+19, 82)-Net over F16 — Constructive and digital
Digital (15, 34, 82)-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 (6, 25, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- digital (0, 9, 17)-net over F16, using
(15, 15+19, 108)-Net over F16 — Digital
Digital (15, 34, 108)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(1634, 108, F16, 2, 19) (dual of [(108, 2), 182, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1634, 129, F16, 2, 19) (dual of [(129, 2), 224, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1634, 258, F16, 19) (dual of [258, 224, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(1634, 256, F16, 19) (dual of [256, 222, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 255 = 162−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(1632, 256, F16, 18) (dual of [256, 224, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 255 = 162−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(160, 2, F16, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(1634, 258, F16, 19) (dual of [258, 224, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(1634, 129, F16, 2, 19) (dual of [(129, 2), 224, 20]-NRT-code), using
(15, 15+19, 129)-Net in Base 16 — Constructive
(15, 34, 129)-net in base 16, using
- 1 times m-reduction [i] based on (15, 35, 129)-net in base 16, using
- base change [i] based on digital (0, 20, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 20, 129)-net over F128, using
(15, 15+19, 7186)-Net in Base 16 — Upper bound on s
There is no (15, 34, 7187)-net in base 16, because
- 1 times m-reduction [i] would yield (15, 33, 7187)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 5451 142899 192789 333461 184014 958489 039796 > 1633 [i]