Best Known (40−11, 40, s)-Nets in Base 16
(40−11, 40, 1815)-Net over F16 — Constructive and digital
Digital (29, 40, 1815)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (1, 3, 273)-net over F16, using
- digital (2, 5, 514)-net over F16, using
- s-reduction based on digital (2, 5, 629)-net over F16, using
- net defined by OOA [i] based on linear OOA(165, 629, F16, 3, 3) (dual of [(629, 3), 1882, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(165, 629, F16, 2, 3) (dual of [(629, 2), 1253, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(165, 629, F16, 3, 3) (dual of [(629, 3), 1882, 4]-NRT-code), using
- s-reduction based on digital (2, 5, 629)-net over F16, using
- digital (5, 10, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 5, 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, 5, 257)-net over F256, using
- digital (11, 22, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 11, 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, 11, 257)-net over F256, using
(40−11, 40, 6555)-Net in Base 16 — Constructive
(29, 40, 6555)-net in base 16, using
- base change [i] based on digital (21, 32, 6555)-net over F32, using
- net defined by OOA [i] based on linear OOA(3232, 6555, F32, 11, 11) (dual of [(6555, 11), 72073, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(3232, 32776, F32, 11) (dual of [32776, 32744, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(3231, 32769, F32, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(3225, 32769, F32, 9) (dual of [32769, 32744, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(3232, 32776, F32, 11) (dual of [32776, 32744, 12]-code), using
- net defined by OOA [i] based on linear OOA(3232, 6555, F32, 11, 11) (dual of [(6555, 11), 72073, 12]-NRT-code), using
(40−11, 40, 19791)-Net over F16 — Digital
Digital (29, 40, 19791)-net over F16, using
(40−11, 40, 20451)-Net in Base 16
(29, 40, 20451)-net in base 16, using
- base change [i] based on digital (21, 32, 20451)-net over F32, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3232, 20451, F32, 11) (dual of [20451, 20419, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(3232, 32776, F32, 11) (dual of [32776, 32744, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(3231, 32769, F32, 11) (dual of [32769, 32738, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(3225, 32769, F32, 9) (dual of [32769, 32744, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(321, 7, F32, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3232, 32776, F32, 11) (dual of [32776, 32744, 12]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3232, 20451, F32, 11) (dual of [20451, 20419, 12]-code), using
(40−11, 40, large)-Net in Base 16 — Upper bound on s
There is no (29, 40, large)-net in base 16, because
- 9 times m-reduction [i] would yield (29, 31, large)-net in base 16, but