Best Known (105−14, 105, s)-Nets in Base 16
(105−14, 105, 2418588)-Net over F16 — Constructive and digital
Digital (91, 105, 2418588)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (18, 25, 21846)-net over F16, using
- net defined by OOA [i] based on linear OOA(1625, 21846, F16, 7, 7) (dual of [(21846, 7), 152897, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1625, 65539, F16, 7) (dual of [65539, 65514, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(1625, 65540, F16, 7) (dual of [65540, 65515, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(1625, 65536, F16, 7) (dual of [65536, 65511, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(1621, 65536, F16, 6) (dual of [65536, 65515, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(160, 4, F16, 0) (dual of [4, 4, 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(1625, 65540, F16, 7) (dual of [65540, 65515, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1625, 65539, F16, 7) (dual of [65539, 65514, 8]-code), using
- net defined by OOA [i] based on linear OOA(1625, 21846, F16, 7, 7) (dual of [(21846, 7), 152897, 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 (18, 25, 21846)-net over F16, using
(105−14, 105, large)-Net over F16 — Digital
Digital (91, 105, large)-net over F16, using
- t-expansion [i] based on digital (89, 105, large)-net over F16, using
- 4 times m-reduction [i] based on digital (89, 109, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16109, large, F16, 20) (dual of [large, large−109, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16109, large, F16, 20) (dual of [large, large−109, 21]-code), using
- 4 times m-reduction [i] based on digital (89, 109, large)-net over F16, using
(105−14, 105, large)-Net in Base 16 — Upper bound on s
There is no (91, 105, large)-net in base 16, because
- 12 times m-reduction [i] would yield (91, 93, large)-net in base 16, but