Best Known (23−7, 23, s)-Nets in Base 16
(23−7, 23, 1623)-Net over F16 — Constructive and digital
Digital (16, 23, 1623)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 257)-net over F16, using
- net defined by OOA [i] based on linear OOA(164, 257, F16, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(164, 257, F16, 2, 3) (dual of [(257, 2), 510, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(164, 257, F16, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
- digital (12, 19, 1366)-net over F16, using
- net defined by OOA [i] based on linear OOA(1619, 1366, F16, 7, 7) (dual of [(1366, 7), 9543, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1619, 4099, F16, 7) (dual of [4099, 4080, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(1619, 4096, F16, 7) (dual of [4096, 4077, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- 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(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(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(1619, 4099, F16, 7) (dual of [4099, 4080, 8]-code), using
- net defined by OOA [i] based on linear OOA(1619, 1366, F16, 7, 7) (dual of [(1366, 7), 9543, 8]-NRT-code), using
- digital (1, 4, 257)-net over F16, using
(23−7, 23, 5461)-Net in Base 16 — Constructive
(16, 23, 5461)-net in base 16, using
- net defined by OOA [i] based on OOA(1623, 5461, S16, 7, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(1623, 16384, S16, 7), using
- discarding factors based on OA(1623, 16386, S16, 7), using
- discarding parts of the base [i] based on linear OA(12813, 16386, F128, 7) (dual of [16386, 16373, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(12811, 16384, F128, 6) (dual of [16384, 16373, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(12813, 16386, F128, 7) (dual of [16386, 16373, 8]-code), using
- discarding factors based on OA(1623, 16386, S16, 7), using
- OOA 3-folding and stacking with additional row [i] based on OA(1623, 16384, S16, 7), using
(23−7, 23, 8243)-Net over F16 — Digital
Digital (16, 23, 8243)-net over F16, using
(23−7, 23, large)-Net in Base 16 — Upper bound on s
There is no (16, 23, large)-net in base 16, because
- 5 times m-reduction [i] would yield (16, 18, large)-net in base 16, but