Best Known (56−20, 56, s)-Nets in Base 16
(56−20, 56, 579)-Net over F16 — Constructive and digital
Digital (36, 56, 579)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (6, 16, 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 (20, 40, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 20, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 20, 257)-net over F256, using
- digital (6, 16, 65)-net over F16, using
(56−20, 56, 2397)-Net over F16 — Digital
Digital (36, 56, 2397)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1656, 2397, F16, 20) (dual of [2397, 2341, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(1656, 4103, F16, 20) (dual of [4103, 4047, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(1655, 4096, F16, 20) (dual of [4096, 4041, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(1649, 4096, F16, 18) (dual of [4096, 4047, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(161, 7, F16, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(1656, 4103, F16, 20) (dual of [4103, 4047, 21]-code), using
(56−20, 56, 1670919)-Net in Base 16 — Upper bound on s
There is no (36, 56, 1670920)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 26 959979 628735 949331 197101 626951 888709 973191 075826 754772 800014 017376 > 1656 [i]