Best Known (223−15, 223, s)-Nets in Base 4
(223−15, 223, 5143012)-Net over F4 — Constructive and digital
Digital (208, 223, 5143012)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (44, 51, 349528)-net over F4, using
- net defined by OOA [i] based on linear OOA(451, 349528, F4, 7, 7) (dual of [(349528, 7), 2446645, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(451, 1048585, F4, 7) (dual of [1048585, 1048534, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(451, 1048586, F4, 7) (dual of [1048586, 1048535, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [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(441, 1048576, F4, 6) (dual of [1048576, 1048535, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 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(451, 1048586, F4, 7) (dual of [1048586, 1048535, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(451, 1048585, F4, 7) (dual of [1048585, 1048534, 8]-code), using
- net defined by OOA [i] based on linear OOA(451, 349528, F4, 7, 7) (dual of [(349528, 7), 2446645, 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 (44, 51, 349528)-net over F4, using
(223−15, 223, large)-Net over F4 — Digital
Digital (208, 223, large)-net over F4, using
- t-expansion [i] based on digital (204, 223, large)-net over F4, using
- 4 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
- 4 times m-reduction [i] based on digital (204, 227, large)-net over F4, using
(223−15, 223, large)-Net in Base 4 — Upper bound on s
There is no (208, 223, large)-net in base 4, because
- 13 times m-reduction [i] would yield (208, 210, large)-net in base 4, but