Best Known (31, 49, s)-Nets in Base 16
(31, 49, 559)-Net over F16 — Constructive and digital
Digital (31, 49, 559)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- digital (18, 36, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 18, 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, 18, 257)-net over F256, using
- digital (4, 13, 45)-net over F16, using
(31, 49, 2049)-Net over F16 — Digital
Digital (31, 49, 2049)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(1649, 2049, F16, 2, 18) (dual of [(2049, 2), 4049, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1649, 4098, F16, 18) (dual of [4098, 4049, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(1649, 4099, F16, 18) (dual of [4099, 4050, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 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(1646, 4096, F16, 17) (dual of [4096, 4050, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(1649, 4099, F16, 18) (dual of [4099, 4050, 19]-code), using
- OOA 2-folding [i] based on linear OA(1649, 4098, F16, 18) (dual of [4098, 4049, 19]-code), using
(31, 49, 994082)-Net in Base 16 — Upper bound on s
There is no (31, 49, 994083)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 100433 872301 234999 735222 135245 293048 146915 790863 917741 514056 > 1649 [i]