Best Known (41, 41+21, s)-Nets in Base 16
(41, 41+21, 1028)-Net over F16 — Constructive and digital
Digital (41, 62, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (10, 20, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 10, 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, 10, 257)-net over F256, using
- digital (21, 42, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 21, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 21, 257)-net over F256, using
- digital (10, 20, 514)-net over F16, using
(41, 41+21, 3872)-Net over F16 — Digital
Digital (41, 62, 3872)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1662, 3872, F16, 21) (dual of [3872, 3810, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1662, 4104, F16, 21) (dual of [4104, 4042, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(1661, 4097, F16, 21) (dual of [4097, 4036, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(1655, 4097, F16, 19) (dual of [4097, 4042, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(1662, 4104, F16, 21) (dual of [4104, 4042, 22]-code), using
(41, 41+21, 6683694)-Net in Base 16 — Upper bound on s
There is no (41, 62, 6683695)-net in base 16, because
- 1 times m-reduction [i] would yield (41, 61, 6683695)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 28 269566 451728 816040 511355 374450 840010 422306 748117 920678 894605 295614 880501 > 1661 [i]