Best Known (117−15, 117, s)-Nets in Base 16
(117−15, 117, 2746268)-Net over F16 — Constructive and digital
Digital (102, 117, 2746268)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (24, 31, 349526)-net over F16, using
- net defined by OOA [i] based on linear OOA(1631, 349526, F16, 7, 7) (dual of [(349526, 7), 2446651, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1631, 1048579, F16, 7) (dual of [1048579, 1048548, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(1631, 1048581, F16, 7) (dual of [1048581, 1048550, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- 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(1626, 1048576, F16, 6) (dual of [1048576, 1048550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 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
- discarding factors / shortening the dual code based on linear OA(1631, 1048581, F16, 7) (dual of [1048581, 1048550, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1631, 1048579, F16, 7) (dual of [1048579, 1048548, 8]-code), using
- net defined by OOA [i] based on linear OOA(1631, 349526, F16, 7, 7) (dual of [(349526, 7), 2446651, 8]-NRT-code), using
- digital (71, 86, 2396742)-net over F16, using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- digital (24, 31, 349526)-net over F16, using
(117−15, 117, large)-Net over F16 — Digital
Digital (102, 117, large)-net over F16, using
- t-expansion [i] based on digital (99, 117, large)-net over F16, using
- 4 times m-reduction [i] based on digital (99, 121, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16121, large, F16, 22) (dual of [large, large−121, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16121, large, F16, 22) (dual of [large, large−121, 23]-code), using
- 4 times m-reduction [i] based on digital (99, 121, large)-net over F16, using
(117−15, 117, large)-Net in Base 16 — Upper bound on s
There is no (102, 117, large)-net in base 16, because
- 13 times m-reduction [i] would yield (102, 104, large)-net in base 16, but