Best Known (204−14, 204, s)-Nets in Base 4
(204−14, 204, 4815336)-Net over F4 — Constructive and digital
Digital (190, 204, 4815336)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (37, 44, 21852)-net over F4, using
- net defined by OOA [i] based on linear OOA(444, 21852, F4, 9, 7) (dual of [(21852, 9), 196624, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(444, 21853, F4, 3, 7) (dual of [(21853, 3), 65515, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(43, 5, F4, 3, 3) (dual of [(5, 3), 12, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;12,4) [i]
- linear OOA(441, 21848, F4, 3, 7) (dual of [(21848, 3), 65503, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(441, 65544, F4, 7) (dual of [65544, 65503, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(433, 65536, F4, 6) (dual of [65536, 65503, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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
- OOA 3-folding [i] based on linear OA(441, 65544, F4, 7) (dual of [65544, 65503, 8]-code), using
- linear OOA(43, 5, F4, 3, 3) (dual of [(5, 3), 12, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(444, 21853, F4, 3, 7) (dual of [(21853, 3), 65515, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(444, 21852, F4, 9, 7) (dual of [(21852, 9), 196624, 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 (37, 44, 21852)-net over F4, using
(204−14, 204, large)-Net over F4 — Digital
Digital (190, 204, large)-net over F4, using
- t-expansion [i] based on digital (186, 204, large)-net over F4, using
- 3 times m-reduction [i] based on digital (186, 207, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4207, large, F4, 21) (dual of [large, large−207, 22]-code), using
- 26 times code embedding in larger space [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 26 times code embedding in larger space [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4207, large, F4, 21) (dual of [large, large−207, 22]-code), using
- 3 times m-reduction [i] based on digital (186, 207, large)-net over F4, using
(204−14, 204, large)-Net in Base 4 — Upper bound on s
There is no (190, 204, large)-net in base 4, because
- 12 times m-reduction [i] would yield (190, 192, large)-net in base 4, but