Best Known (79−11, 79, s)-Nets in Base 16
(79−11, 79, 3420993)-Net over F16 — Constructive and digital
Digital (68, 79, 3420993)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (12, 17, 65553)-net over F16, using
- net defined by OOA [i] based on linear OOA(1617, 65553, F16, 6, 5) (dual of [(65553, 6), 393301, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(1617, 65554, F16, 2, 5) (dual of [(65554, 2), 131091, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(163, 273, F16, 2, 2) (dual of [(273, 2), 543, 3]-NRT-code), using
- appending kth column [i] based on linear OA(163, 273, F16, 2) (dual of [273, 270, 3]-code), using
- Hamming code H(3,16) [i]
- appending kth column [i] based on linear OA(163, 273, F16, 2) (dual of [273, 270, 3]-code), using
- linear OOA(1614, 65281, F16, 2, 5) (dual of [(65281, 2), 130548, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- trace code [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- OOA 2-folding [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- linear OOA(163, 273, F16, 2, 2) (dual of [(273, 2), 543, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(1617, 65554, F16, 2, 5) (dual of [(65554, 2), 131091, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1617, 65553, F16, 6, 5) (dual of [(65553, 6), 393301, 6]-NRT-code), using
- digital (51, 62, 3355440)-net over F16, using
- trace code for nets [i] based on digital (20, 31, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- trace code for nets [i] based on digital (20, 31, 1677720)-net over F256, using
- digital (12, 17, 65553)-net over F16, using
(79−11, 79, large)-Net over F16 — Digital
Digital (68, 79, large)-net over F16, using
- t-expansion [i] based on digital (65, 79, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1679, large, F16, 14) (dual of [large, large−79, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1679, large, F16, 14) (dual of [large, large−79, 15]-code), using
(79−11, 79, large)-Net in Base 16 — Upper bound on s
There is no (68, 79, large)-net in base 16, because
- 9 times m-reduction [i] would yield (68, 70, large)-net in base 16, but