Best Known (231−14, 231, s)-Nets in Base 4
(231−14, 231, 7594724)-Net over F4 — Constructive and digital
Digital (217, 231, 7594724)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (64, 71, 2801240)-net over F4, using
- net defined by OOA [i] based on linear OOA(471, 2801240, F4, 9, 7) (dual of [(2801240, 9), 25211089, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(471, 2801241, F4, 3, 7) (dual of [(2801241, 3), 8403652, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(410, 5040, F4, 3, 3) (dual of [(5040, 3), 15110, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(410, 5040, F4, 2, 3) (dual of [(5040, 2), 10070, 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(410, 5040, F4, 3, 3) (dual of [(5040, 3), 15110, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(471, 2801241, F4, 3, 7) (dual of [(2801241, 3), 8403652, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(471, 2801240, F4, 9, 7) (dual of [(2801240, 9), 25211089, 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 (64, 71, 2801240)-net over F4, using
(231−14, 231, large)-Net over F4 — Digital
Digital (217, 231, large)-net over F4, using
- t-expansion [i] based on digital (213, 231, large)-net over F4, using
- 6 times m-reduction [i] based on digital (213, 237, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4237, large, F4, 24) (dual of [large, large−237, 25]-code), using
- 21 times code embedding in larger space [i] based on linear OA(4216, large, F4, 24) (dual of [large, large−216, 25]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- 21 times code embedding in larger space [i] based on linear OA(4216, large, F4, 24) (dual of [large, large−216, 25]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4237, large, F4, 24) (dual of [large, large−237, 25]-code), using
- 6 times m-reduction [i] based on digital (213, 237, large)-net over F4, using
(231−14, 231, large)-Net in Base 4 — Upper bound on s
There is no (217, 231, large)-net in base 4, because
- 12 times m-reduction [i] would yield (217, 219, large)-net in base 4, but