Best Known (17, 17+6, s)-Nets in Base 16
(17, 17+6, 43692)-Net over F16 — Constructive and digital
Digital (17, 23, 43692)-net over F16, using
- 161 times duplication [i] based on digital (16, 22, 43692)-net over F16, using
- net defined by OOA [i] based on linear OOA(1622, 43692, F16, 6, 6) (dual of [(43692, 6), 262130, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1622, 131076, F16, 6) (dual of [131076, 131054, 7]-code), using
- trace code [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2569, 65536, F256, 5) (dual of [65536, 65527, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- trace code [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(1622, 131076, F16, 6) (dual of [131076, 131054, 7]-code), using
- net defined by OOA [i] based on linear OOA(1622, 43692, F16, 6, 6) (dual of [(43692, 6), 262130, 7]-NRT-code), using
(17, 17+6, 131078)-Net over F16 — Digital
Digital (17, 23, 131078)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1623, 131078, F16, 6) (dual of [131078, 131055, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1622, 131076, F16, 6) (dual of [131076, 131054, 7]-code), using
- trace code [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2569, 65536, F256, 5) (dual of [65536, 65527, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- trace code [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- linear OA(1622, 131077, F16, 5) (dual of [131077, 131055, 6]-code), using Gilbert–Varšamov bound and bm = 1622 > Vbs−1(k−1) = 622626 379384 691686 998016 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 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
- linear OA(1622, 131076, F16, 6) (dual of [131076, 131054, 7]-code), using
- construction X with Varšamov bound [i] based on
(17, 17+6, large)-Net in Base 16 — Upper bound on s
There is no (17, 23, large)-net in base 16, because
- 4 times m-reduction [i] would yield (17, 19, large)-net in base 16, but