Best Known (99, 113, s)-Nets in Base 16
(99, 113, 2746272)-Net over F16 — Constructive and digital
Digital (99, 113, 2746272)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (26, 33, 349530)-net over F16, using
- net defined by OOA [i] based on linear OOA(1633, 349530, F16, 7, 7) (dual of [(349530, 7), 2446677, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1633, 1048591, F16, 7) (dual of [1048591, 1048558, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(1633, 1048593, F16, 7) (dual of [1048593, 1048560, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [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(1616, 1048576, F16, 4) (dual of [1048576, 1048560, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(162, 17, F16, 2) (dual of [17, 15, 3]-code or 17-arc in PG(1,16)), using
- extended Reed–Solomon code RSe(15,16) [i]
- Hamming code H(2,16) [i]
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(1633, 1048593, F16, 7) (dual of [1048593, 1048560, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1633, 1048591, F16, 7) (dual of [1048591, 1048558, 8]-code), using
- net defined by OOA [i] based on linear OOA(1633, 349530, F16, 7, 7) (dual of [(349530, 7), 2446677, 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 (26, 33, 349530)-net over F16, using
(99, 113, large)-Net over F16 — Digital
Digital (99, 113, large)-net over F16, using
- 8 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
(99, 113, large)-Net in Base 16 — Upper bound on s
There is no (99, 113, large)-net in base 16, because
- 12 times m-reduction [i] would yield (99, 101, large)-net in base 16, but