Best Known (28−9, 28, s)-Nets in Base 16
(28−9, 28, 1030)-Net over F16 — Constructive and digital
Digital (19, 28, 1030)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (4, 8, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 4, 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, 4, 257)-net over F256, using
- digital (11, 20, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 10, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 10, 258)-net over F256, using
- digital (4, 8, 514)-net over F16, using
(28−9, 28, 4481)-Net over F16 — Digital
Digital (19, 28, 4481)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1628, 4481, F16, 9) (dual of [4481, 4453, 10]-code), using
- 375 step Varšamov–Edel lengthening with (ri) = (1, 28 times 0, 1, 345 times 0) [i] based on linear OA(1626, 4104, F16, 9) (dual of [4104, 4078, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(1625, 4097, F16, 9) (dual of [4097, 4072, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(1619, 4097, F16, 7) (dual of [4097, 4078, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,4]) ⊂ C([0,3]) [i] based on
- 375 step Varšamov–Edel lengthening with (ri) = (1, 28 times 0, 1, 345 times 0) [i] based on linear OA(1626, 4104, F16, 9) (dual of [4104, 4078, 10]-code), using
(28−9, 28, large)-Net in Base 16 — Upper bound on s
There is no (19, 28, large)-net in base 16, because
- 7 times m-reduction [i] would yield (19, 21, large)-net in base 16, but