Best Known (143−10, 143, s)-Nets in Base 4
(143−10, 143, 7235167)-Net over F4 — Constructive and digital
Digital (133, 143, 7235167)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (26, 31, 524287)-net over F4, using
- net defined by OOA [i] based on linear OOA(431, 524287, F4, 5, 5) (dual of [(524287, 5), 2621404, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(431, 1048575, F4, 5) (dual of [1048575, 1048544, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(431, 1048576, F4, 5) (dual of [1048576, 1048545, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(431, 1048576, F4, 5) (dual of [1048576, 1048545, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(431, 1048575, F4, 5) (dual of [1048575, 1048544, 6]-code), using
- net defined by OOA [i] based on linear OOA(431, 524287, F4, 5, 5) (dual of [(524287, 5), 2621404, 6]-NRT-code), using
- digital (102, 112, 6710880)-net over F4, using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(25628, large, F256, 10) (dual of [large, large−28, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(25628, 8388600, F256, 10) (dual of [8388600, 8388572, 11]-code), using
- net defined by OOA [i] based on linear OOA(25628, 1677720, F256, 10, 10) (dual of [(1677720, 10), 16777172, 11]-NRT-code), using
- trace code for nets [i] based on digital (18, 28, 1677720)-net over F256, using
- digital (26, 31, 524287)-net over F4, using
(143−10, 143, large)-Net over F4 — Digital
Digital (133, 143, large)-net over F4, using
- t-expansion [i] based on digital (130, 143, large)-net over F4, using
- 2 times m-reduction [i] based on digital (130, 145, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4145, large, F4, 15) (dual of [large, large−145, 16]-code), using
- strength reduction [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- strength reduction [i] based on linear OA(4145, large, F4, 17) (dual of [large, large−145, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4145, large, F4, 15) (dual of [large, large−145, 16]-code), using
- 2 times m-reduction [i] based on digital (130, 145, large)-net over F4, using
(143−10, 143, large)-Net in Base 4 — Upper bound on s
There is no (133, 143, large)-net in base 4, because
- 8 times m-reduction [i] would yield (133, 135, large)-net in base 4, but