Best Known (253, 253+3, s)-Nets in Base 4
(253, 253+3, large)-Net over F4 — Constructive and digital
Digital (253, 256, large)-net over F4, using
- 44 times duplication [i] based on digital (249, 252, large)-net over F4, using
- t-expansion [i] based on digital (238, 252, large)-net over F4, using
- trace code for nets [i] based on digital (49, 63, 2097278)-net over F256, using
- net defined by OOA [i] based on linear OOA(25663, 2097278, F256, 20, 14) (dual of [(2097278, 20), 41945497, 15]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(25663, 4194557, F256, 4, 14) (dual of [(4194557, 4), 16778165, 15]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(2564, 257, F256, 4, 4) (dual of [(257, 4), 1024, 5]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(4;1024,256) [i]
- linear OOA(25619, 2097150, F256, 4, 7) (dual of [(2097150, 4), 8388581, 8]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25619, 2796200, F256, 4, 7) (dual of [(2796200, 4), 11184781, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25619, 8388601, F256, 7) (dual of [8388601, 8388582, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25619, 8388601, F256, 7) (dual of [8388601, 8388582, 8]-code), using
- discarding factors / shortening the dual code based on linear OOA(25619, 2796200, F256, 4, 7) (dual of [(2796200, 4), 11184781, 8]-NRT-code), using
- linear OOA(25640, 2097150, F256, 4, 14) (dual of [(2097150, 4), 8388560, 15]-NRT-code), using
- OOA 4-folding [i] based on linear OA(25640, 8388600, F256, 14) (dual of [8388600, 8388560, 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
- OOA 4-folding [i] based on linear OA(25640, 8388600, F256, 14) (dual of [8388600, 8388560, 15]-code), using
- linear OOA(2564, 257, F256, 4, 4) (dual of [(257, 4), 1024, 5]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OOA(25663, 4194557, F256, 4, 14) (dual of [(4194557, 4), 16778165, 15]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25663, 2097278, F256, 20, 14) (dual of [(2097278, 20), 41945497, 15]-NRT-code), using
- trace code for nets [i] based on digital (49, 63, 2097278)-net over F256, using
- t-expansion [i] based on digital (238, 252, large)-net over F4, using
(253, 253+3, large)-Net in Base 4 — Upper bound on s
There is no (253, 256, large)-net in base 4, because
- 1 times m-reduction [i] would yield (253, 255, large)-net in base 4, but