Best Known (19, 19+10, s)-Nets in Base 16
(19, 19+10, 820)-Net over F16 — Constructive and digital
Digital (19, 29, 820)-net over F16, using
- net defined by OOA [i] based on linear OOA(1629, 820, F16, 10, 10) (dual of [(820, 10), 8171, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(1629, 4100, F16, 10) (dual of [4100, 4071, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1629, 4103, F16, 10) (dual of [4103, 4074, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(1628, 4096, F16, 10) (dual of [4096, 4068, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1622, 4096, F16, 8) (dual of [4096, 4074, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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 Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(1629, 4103, F16, 10) (dual of [4103, 4074, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(1629, 4100, F16, 10) (dual of [4100, 4071, 11]-code), using
(19, 19+10, 1010)-Net in Base 16 — Constructive
(19, 29, 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 (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
- (4, 9, 496)-net in base 16, using
(19, 19+10, 4103)-Net over F16 — Digital
Digital (19, 29, 4103)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1629, 4103, F16, 10) (dual of [4103, 4074, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(1628, 4096, F16, 10) (dual of [4096, 4068, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1622, 4096, F16, 8) (dual of [4096, 4074, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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 Ce(9) ⊂ Ce(7) [i] based on
(19, 19+10, 1673556)-Net in Base 16 — Upper bound on s
There is no (19, 29, 1673557)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 83076 964571 427646 553461 664240 339776 > 1629 [i]