Best Known (211, 226, s)-Nets in Base 4
(211, 226, 5143017)-Net over F4 — Constructive and digital
Digital (211, 226, 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 (157, 172, 4793484)-net over F4, using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- digital (47, 54, 349533)-net over F4, using
(211, 226, large)-Net over F4 — Digital
Digital (211, 226, large)-net over F4, using
- t-expansion [i] based on digital (204, 226, large)-net over F4, using
- 1 times m-reduction [i] based on digital (204, 227, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4227, large, F4, 23) (dual of [large, large−227, 24]-code), using
- 22 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 22 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4227, large, F4, 23) (dual of [large, large−227, 24]-code), using
- 1 times m-reduction [i] based on digital (204, 227, large)-net over F4, using
(211, 226, large)-Net in Base 4 — Upper bound on s
There is no (211, 226, large)-net in base 4, because
- 13 times m-reduction [i] would yield (211, 213, large)-net in base 4, but