Best Known (67, 67+22, s)-Nets in Base 16
(67, 67+22, 11916)-Net over F16 — Constructive and digital
Digital (67, 89, 11916)-net over F16, using
- 163 times duplication [i] based on digital (64, 86, 11916)-net over F16, using
- net defined by OOA [i] based on linear OOA(1686, 11916, F16, 22, 22) (dual of [(11916, 22), 262066, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(1686, 131076, F16, 22) (dual of [131076, 130990, 23]-code), using
- trace code [i] based on linear OA(25643, 65538, F256, 22) (dual of [65538, 65495, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(25643, 65536, F256, 22) (dual of [65536, 65493, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- trace code [i] based on linear OA(25643, 65538, F256, 22) (dual of [65538, 65495, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(1686, 131076, F16, 22) (dual of [131076, 130990, 23]-code), using
- net defined by OOA [i] based on linear OOA(1686, 11916, F16, 22, 22) (dual of [(11916, 22), 262066, 23]-NRT-code), using
(67, 67+22, 109977)-Net over F16 — Digital
Digital (67, 89, 109977)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1689, 109977, F16, 22) (dual of [109977, 109888, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(1689, 131083, F16, 22) (dual of [131083, 130994, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1688, 131082, F16, 22) (dual of [131082, 130994, 23]-code), using
- trace code [i] based on linear OA(25644, 65541, F256, 22) (dual of [65541, 65497, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(25643, 65536, F256, 22) (dual of [65536, 65493, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(25639, 65536, F256, 20) (dual of [65536, 65497, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- trace code [i] based on linear OA(25644, 65541, F256, 22) (dual of [65541, 65497, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1688, 131082, F16, 22) (dual of [131082, 130994, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(1689, 131083, F16, 22) (dual of [131083, 130994, 23]-code), using
(67, 67+22, large)-Net in Base 16 — Upper bound on s
There is no (67, 89, large)-net in base 16, because
- 20 times m-reduction [i] would yield (67, 69, large)-net in base 16, but