Best Known (134−10, 134, s)-Nets in Base 4
(134−10, 134, 6719071)-Net over F4 — Constructive and digital
Digital (124, 134, 6719071)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (17, 22, 8191)-net over F4, using
- net defined by OOA [i] based on linear OOA(422, 8191, F4, 5, 5) (dual of [(8191, 5), 40933, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(422, 16383, F4, 5) (dual of [16383, 16361, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(422, 16383, F4, 5) (dual of [16383, 16361, 6]-code), using
- net defined by OOA [i] based on linear OOA(422, 8191, F4, 5, 5) (dual of [(8191, 5), 40933, 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 (17, 22, 8191)-net over F4, using
(134−10, 134, large)-Net over F4 — Digital
Digital (124, 134, large)-net over F4, using
- t-expansion [i] based on digital (121, 134, large)-net over F4, using
- 1 times m-reduction [i] based on digital (121, 135, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4135, large, F4, 14) (dual of [large, large−135, 15]-code), using
- 14 times code embedding in larger space [i] based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 14 times code embedding in larger space [i] based on linear OA(4121, large, F4, 14) (dual of [large, large−121, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4135, large, F4, 14) (dual of [large, large−135, 15]-code), using
- 1 times m-reduction [i] based on digital (121, 135, large)-net over F4, using
(134−10, 134, large)-Net in Base 4 — Upper bound on s
There is no (124, 134, large)-net in base 4, because
- 8 times m-reduction [i] would yield (124, 126, large)-net in base 4, but