Best Known (214−14, 214, s)-Nets in Base 4
(214−14, 214, 5143017)-Net over F4 — Constructive and digital
Digital (200, 214, 5143017)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (47, 54, 349533)-net over F4, using
- net defined by OOA [i] based on linear OOA(454, 349533, F4, 7, 7) (dual of [(349533, 7), 2446677, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(454, 1048600, F4, 7) (dual of [1048600, 1048546, 8]-code), using
- 2 times code embedding in larger space [i] based on linear OA(452, 1048598, F4, 7) (dual of [1048598, 1048546, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(451, 1048576, F4, 7) (dual of [1048576, 1048525, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(431, 1048576, F4, 5) (dual of [1048576, 1048545, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(421, 22, F4, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,4)), using
- dual of repetition code with length 22 [i]
- linear OA(41, 22, F4, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(452, 1048598, F4, 7) (dual of [1048598, 1048546, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(454, 1048600, F4, 7) (dual of [1048600, 1048546, 8]-code), using
- net defined by OOA [i] based on linear OOA(454, 349533, F4, 7, 7) (dual of [(349533, 7), 2446677, 8]-NRT-code), using
- digital (146, 160, 4793484)-net over F4, 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 (47, 54, 349533)-net over F4, using
(214−14, 214, large)-Net over F4 — Digital
Digital (200, 214, large)-net over F4, using
- t-expansion [i] based on digital (195, 214, large)-net over F4, using
- 3 times m-reduction [i] based on digital (195, 217, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4217, large, F4, 22) (dual of [large, large−217, 23]-code), using
- strength reduction [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- strength reduction [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4217, large, F4, 22) (dual of [large, large−217, 23]-code), using
- 3 times m-reduction [i] based on digital (195, 217, large)-net over F4, using
(214−14, 214, large)-Net in Base 4 — Upper bound on s
There is no (200, 214, large)-net in base 4, because
- 12 times m-reduction [i] would yield (200, 202, large)-net in base 4, but