Best Known (230, 230+15, s)-Nets in Base 4
(230, 230+15, 7624250)-Net over F4 — Constructive and digital
Digital (230, 245, 7624250)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (66, 73, 2830766)-net over F4, using
- net defined by OOA [i] based on linear OOA(473, 2830766, F4, 9, 7) (dual of [(2830766, 9), 25476821, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(473, 2830767, F4, 3, 7) (dual of [(2830767, 3), 8492228, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(412, 34566, F4, 3, 3) (dual of [(34566, 3), 103686, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(412, 34566, F4, 2, 3) (dual of [(34566, 2), 69120, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(412, 34566, F4, 3) (dual of [34566, 34554, 4]-code or 34566-cap in PG(11,4)), using
- appending kth column [i] based on linear OOA(412, 34566, F4, 2, 3) (dual of [(34566, 2), 69120, 4]-NRT-code), using
- linear OOA(461, 2796201, F4, 3, 7) (dual of [(2796201, 3), 8388542, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- OOA 3-folding [i] based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- linear OOA(412, 34566, F4, 3, 3) (dual of [(34566, 3), 103686, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(473, 2830767, F4, 3, 7) (dual of [(2830767, 3), 8492228, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(473, 2830766, F4, 9, 7) (dual of [(2830766, 9), 25476821, 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 (66, 73, 2830766)-net over F4, using
(230, 230+15, large)-Net over F4 — Digital
Digital (230, 245, large)-net over F4, using
- t-expansion [i] based on digital (222, 245, large)-net over F4, using
- 2 times m-reduction [i] based on digital (222, 247, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
- 30 times code embedding in larger space [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]
- 30 times code embedding in larger space [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(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
- 2 times m-reduction [i] based on digital (222, 247, large)-net over F4, using
(230, 230+15, large)-Net in Base 4 — Upper bound on s
There is no (230, 245, large)-net in base 4, because
- 13 times m-reduction [i] would yield (230, 232, large)-net in base 4, but