Best Known (102, 116, s)-Nets in Base 16
(102, 116, 2746897)-Net over F16 — Constructive and digital
Digital (102, 116, 2746897)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (29, 36, 350155)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (2, 5, 629)-net over F16, using
- net defined by OOA [i] based on linear OOA(165, 629, F16, 3, 3) (dual of [(629, 3), 1882, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(165, 629, F16, 2, 3) (dual of [(629, 2), 1253, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(165, 629, F16, 3, 3) (dual of [(629, 3), 1882, 4]-NRT-code), using
- 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 (2, 5, 629)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (66, 80, 2396742)-net over F16, using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- digital (29, 36, 350155)-net over F16, using
(102, 116, 3095795)-Net in Base 16 — Constructive
(102, 116, 3095795)-net in base 16, using
- 161 times duplication [i] based on (101, 115, 3095795)-net in base 16, using
- (u, u+v)-construction [i] based on
- (28, 35, 699053)-net in base 16, using
- base change [i] based on digital (13, 20, 699053)-net over F128, using
- net defined by OOA [i] based on linear OOA(12820, 699053, F128, 7, 7) (dual of [(699053, 7), 4893351, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12820, 2097160, F128, 7) (dual of [2097160, 2097140, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(12819, 2097153, F128, 7) (dual of [2097153, 2097134, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(12813, 2097153, F128, 5) (dual of [2097153, 2097140, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 1286−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(1281, 7, F128, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(12820, 2097160, F128, 7) (dual of [2097160, 2097140, 8]-code), using
- net defined by OOA [i] based on linear OOA(12820, 699053, F128, 7, 7) (dual of [(699053, 7), 4893351, 8]-NRT-code), using
- base change [i] based on digital (13, 20, 699053)-net over F128, using
- digital (66, 80, 2396742)-net over F16, using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- (28, 35, 699053)-net in base 16, using
- (u, u+v)-construction [i] based on
(102, 116, large)-Net over F16 — Digital
Digital (102, 116, large)-net over F16, using
- t-expansion [i] based on digital (99, 116, large)-net over F16, using
- 5 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
- 5 times m-reduction [i] based on digital (99, 121, large)-net over F16, using
(102, 116, large)-Net in Base 16 — Upper bound on s
There is no (102, 116, large)-net in base 16, because
- 12 times m-reduction [i] would yield (102, 104, large)-net in base 16, but