Best Known (112−14, 112, s)-Nets in Base 16
(112−14, 112, 2746271)-Net over F16 — Constructive and digital
Digital (98, 112, 2746271)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (25, 32, 349529)-net over F16, using
- net defined by OOA [i] based on linear OOA(1632, 349529, F16, 7, 7) (dual of [(349529, 7), 2446671, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1632, 1048588, F16, 7) (dual of [1048588, 1048556, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(1631, 1048577, F16, 7) (dual of [1048577, 1048546, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(1621, 1048577, F16, 5) (dual of [1048577, 1048556, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 1610−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(161, 11, F16, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 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(1632, 1048588, F16, 7) (dual of [1048588, 1048556, 8]-code), using
- net defined by OOA [i] based on linear OOA(1632, 349529, F16, 7, 7) (dual of [(349529, 7), 2446671, 8]-NRT-code), 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
- digital (25, 32, 349529)-net over F16, using
(112−14, 112, large)-Net over F16 — Digital
Digital (98, 112, large)-net over F16, using
- t-expansion [i] based on digital (94, 112, large)-net over F16, using
- 3 times m-reduction [i] based on digital (94, 115, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16115, large, F16, 21) (dual of [large, large−115, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16115, large, F16, 21) (dual of [large, large−115, 22]-code), using
- 3 times m-reduction [i] based on digital (94, 115, large)-net over F16, using
(112−14, 112, large)-Net in Base 16 — Upper bound on s
There is no (98, 112, large)-net in base 16, because
- 12 times m-reduction [i] would yield (98, 100, large)-net in base 16, but