Best Known (13, 19, s)-Nets in Base 16
(13, 19, 1383)-Net over F16 — Constructive and digital
Digital (13, 19, 1383)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 17)-net over F16, using
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 0 and N(F) ≥ 17, using
- the rational function field F16(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 16)-sequence over F16, using
- digital (10, 16, 1366)-net over F16, using
- net defined by OOA [i] based on linear OOA(1616, 1366, F16, 6, 6) (dual of [(1366, 6), 8180, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1616, 4098, F16, 6) (dual of [4098, 4082, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(1616, 4099, F16, 6) (dual of [4099, 4083, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(1616, 4096, F16, 6) (dual of [4096, 4080, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1613, 4096, F16, 5) (dual of [4096, 4083, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(1616, 4099, F16, 6) (dual of [4099, 4083, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(1616, 4098, F16, 6) (dual of [4098, 4082, 7]-code), using
- net defined by OOA [i] based on linear OOA(1616, 1366, F16, 6, 6) (dual of [(1366, 6), 8180, 7]-NRT-code), using
- digital (0, 3, 17)-net over F16, using
(13, 19, 6661)-Net over F16 — Digital
Digital (13, 19, 6661)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1619, 6661, F16, 6) (dual of [6661, 6642, 7]-code), using
- 2559 step Varšamov–Edel lengthening with (ri) = (1, 29 times 0, 1, 396 times 0, 1, 2131 times 0) [i] based on linear OA(1616, 4099, F16, 6) (dual of [4099, 4083, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(1616, 4096, F16, 6) (dual of [4096, 4080, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1613, 4096, F16, 5) (dual of [4096, 4083, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- 2559 step Varšamov–Edel lengthening with (ri) = (1, 29 times 0, 1, 396 times 0, 1, 2131 times 0) [i] based on linear OA(1616, 4099, F16, 6) (dual of [4099, 4083, 7]-code), using
(13, 19, 5121363)-Net in Base 16 — Upper bound on s
There is no (13, 19, 5121364)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 75557 880183 768297 516631 > 1619 [i]