Best Known (143−11, 143, s)-Nets in Base 4
(143−11, 143, 6712927)-Net over F4 — Constructive and digital
Digital (132, 143, 6712927)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (14, 19, 2047)-net over F4, using
- net defined by OOA [i] based on linear OOA(419, 2047, F4, 5, 5) (dual of [(2047, 5), 10216, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(419, 4095, F4, 5) (dual of [4095, 4076, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(419, 4096, F4, 5) (dual of [4096, 4077, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(419, 4095, F4, 5) (dual of [4095, 4076, 6]-code), using
- net defined by OOA [i] based on linear OOA(419, 2047, F4, 5, 5) (dual of [(2047, 5), 10216, 6]-NRT-code), using
- digital (113, 124, 6710880)-net over F4, using
- trace code for nets [i] based on digital (20, 31, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- trace code for nets [i] based on digital (20, 31, 1677720)-net over F256, using
- digital (14, 19, 2047)-net over F4, using
(143−11, 143, large)-Net over F4 — Digital
Digital (132, 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
(143−11, 143, large)-Net in Base 4 — Upper bound on s
There is no (132, 143, large)-net in base 4, because
- 9 times m-reduction [i] would yield (132, 134, large)-net in base 4, but