Best Known (31−11, 31, s)-Nets in Base 16
(31−11, 31, 819)-Net over F16 — Constructive and digital
Digital (20, 31, 819)-net over F16, using
- net defined by OOA [i] based on linear OOA(1631, 819, F16, 11, 11) (dual of [(819, 11), 8978, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(1631, 4096, F16, 11) (dual of [4096, 4065, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- OOA 5-folding and stacking with additional row [i] based on linear OA(1631, 4096, F16, 11) (dual of [4096, 4065, 12]-code), using
(31−11, 31, 1010)-Net in Base 16 — Constructive
(20, 31, 1010)-net in base 16, using
- (u, u+v)-construction [i] based on
- (4, 9, 496)-net in base 16, using
- net defined by OOA [i] based on OOA(169, 496, S16, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(169, 993, S16, 5), using
- discarding parts of the base [i] based on linear OA(327, 993, F32, 5) (dual of [993, 986, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on OA(169, 993, S16, 5), using
- net defined by OOA [i] based on OOA(169, 496, S16, 5, 5), 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, 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, 11, 257)-net over F256, using
- (4, 9, 496)-net in base 16, using
(31−11, 31, 2850)-Net over F16 — Digital
Digital (20, 31, 2850)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1631, 2850, F16, 11) (dual of [2850, 2819, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(1631, 4096, F16, 11) (dual of [4096, 4065, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(1631, 4096, F16, 11) (dual of [4096, 4065, 12]-code), using
(31−11, 31, 2913832)-Net in Base 16 — Upper bound on s
There is no (20, 31, 2913833)-net in base 16, because
- 1 times m-reduction [i] would yield (20, 30, 2913833)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 1 329229 819749 255321 824241 899120 355976 > 1630 [i]