Best Known (16, 16+20, s)-Nets in Base 16
(16, 16+20, 82)-Net over F16 — Constructive and digital
Digital (16, 36, 82)-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 (6, 26, 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, 10, 17)-net over F16, using
(16, 16+20, 114)-Net over F16 — Digital
Digital (16, 36, 114)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(1636, 114, F16, 2, 20) (dual of [(114, 2), 192, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1636, 129, F16, 2, 20) (dual of [(129, 2), 222, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1636, 258, F16, 20) (dual of [258, 222, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(1636, 256, F16, 20) (dual of [256, 220, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 255 = 162−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- 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(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(19) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(1636, 258, F16, 20) (dual of [258, 222, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(1636, 129, F16, 2, 20) (dual of [(129, 2), 222, 21]-NRT-code), using
(16, 16+20, 129)-Net in Base 16 — Constructive
(16, 36, 129)-net in base 16, using
- 161 times duplication [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
(16, 16+20, 6521)-Net in Base 16 — Upper bound on s
There is no (16, 36, 6522)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 22 304671 612665 301326 923846 106162 225779 291176 > 1636 [i]