Best Known (33−8, 33, s)-Nets in Base 16
(33−8, 33, 32770)-Net over F16 — Constructive and digital
Digital (25, 33, 32770)-net over F16, using
- 161 times duplication [i] based on digital (24, 32, 32770)-net over F16, using
- net defined by OOA [i] based on linear OOA(1632, 32770, F16, 8, 8) (dual of [(32770, 8), 262128, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(1632, 131080, F16, 8) (dual of [131080, 131048, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(1632, 131082, F16, 8) (dual of [131082, 131050, 9]-code), using
- trace code [i] based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(1632, 131082, F16, 8) (dual of [131082, 131050, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(1632, 131080, F16, 8) (dual of [131080, 131048, 9]-code), using
- net defined by OOA [i] based on linear OOA(1632, 32770, F16, 8, 8) (dual of [(32770, 8), 262128, 9]-NRT-code), using
(33−8, 33, 65536)-Net in Base 16 — Constructive
(25, 33, 65536)-net in base 16, using
- base change [i] based on digital (14, 22, 65536)-net over F64, using
- net defined by OOA [i] based on linear OOA(6422, 65536, F64, 8, 8) (dual of [(65536, 8), 524266, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OA 4-folding and stacking [i] based on linear OA(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using
- net defined by OOA [i] based on linear OOA(6422, 65536, F64, 8, 8) (dual of [(65536, 8), 524266, 9]-NRT-code), using
(33−8, 33, 131084)-Net over F16 — Digital
Digital (25, 33, 131084)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1633, 131084, F16, 8) (dual of [131084, 131051, 9]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1632, 131082, F16, 8) (dual of [131082, 131050, 9]-code), using
- trace code [i] based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 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(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(7) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(25616, 65541, F256, 8) (dual of [65541, 65525, 9]-code), using
- linear OA(1632, 131083, F16, 7) (dual of [131083, 131051, 8]-code), using Gilbert–Varšamov bound and bm = 1632 > Vbs−1(k−1) = 80246 304169 023850 471583 378724 815956 [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(1632, 131082, F16, 8) (dual of [131082, 131050, 9]-code), using
- construction X with Varšamov bound [i] based on
(33−8, 33, large)-Net in Base 16 — Upper bound on s
There is no (25, 33, large)-net in base 16, because
- 6 times m-reduction [i] would yield (25, 27, large)-net in base 16, but