Best Known (49−10, 49, s)-Nets in Base 16
(49−10, 49, 209718)-Net over F16 — Constructive and digital
Digital (39, 49, 209718)-net over F16, using
- 161 times duplication [i] based on digital (38, 48, 209718)-net over F16, using
- net defined by OOA [i] based on linear OOA(1648, 209718, F16, 10, 10) (dual of [(209718, 10), 2097132, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(1648, 1048590, F16, 10) (dual of [1048590, 1048542, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1648, 1048593, F16, 10) (dual of [1048593, 1048545, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(1646, 1048576, F16, 10) (dual of [1048576, 1048530, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1631, 1048576, F16, 7) (dual of [1048576, 1048545, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(162, 17, F16, 2) (dual of [17, 15, 3]-code or 17-arc in PG(1,16)), using
- extended Reed–Solomon code RSe(15,16) [i]
- Hamming code H(2,16) [i]
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(1648, 1048593, F16, 10) (dual of [1048593, 1048545, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(1648, 1048590, F16, 10) (dual of [1048590, 1048542, 11]-code), using
- net defined by OOA [i] based on linear OOA(1648, 209718, F16, 10, 10) (dual of [(209718, 10), 2097132, 11]-NRT-code), using
(49−10, 49, 419431)-Net in Base 16 — Constructive
(39, 49, 419431)-net in base 16, using
- base change [i] based on digital (18, 28, 419431)-net over F128, using
- net defined by OOA [i] based on linear OOA(12828, 419431, F128, 10, 10) (dual of [(419431, 10), 4194282, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(12828, 2097155, F128, 10) (dual of [2097155, 2097127, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(12828, 2097152, F128, 10) (dual of [2097152, 2097124, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(12825, 2097152, F128, 9) (dual of [2097152, 2097127, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 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(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(12828, 2097155, F128, 10) (dual of [2097155, 2097127, 11]-code), using
- net defined by OOA [i] based on linear OOA(12828, 419431, F128, 10, 10) (dual of [(419431, 10), 4194282, 11]-NRT-code), using
(49−10, 49, 1048595)-Net over F16 — Digital
Digital (39, 49, 1048595)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1649, 1048595, F16, 10) (dual of [1048595, 1048546, 11]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1648, 1048593, F16, 10) (dual of [1048593, 1048545, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(1646, 1048576, F16, 10) (dual of [1048576, 1048530, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1631, 1048576, F16, 7) (dual of [1048576, 1048545, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(162, 17, F16, 2) (dual of [17, 15, 3]-code or 17-arc in PG(1,16)), using
- extended Reed–Solomon code RSe(15,16) [i]
- Hamming code H(2,16) [i]
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(1648, 1048594, F16, 9) (dual of [1048594, 1048546, 10]-code), using Gilbert–Varšamov bound and bm = 1648 > Vbs−1(k−1) = 92908 148794 100743 465711 956012 023298 980908 357612 295446 [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(1648, 1048593, F16, 10) (dual of [1048593, 1048545, 11]-code), using
- construction X with Varšamov bound [i] based on
(49−10, 49, large)-Net in Base 16 — Upper bound on s
There is no (39, 49, large)-net in base 16, because
- 8 times m-reduction [i] would yield (39, 41, large)-net in base 16, but