Best Known (123, 123+10, s)-Nets in Base 4
(123, 123+10, 6716929)-Net over F4 — Constructive and digital
Digital (123, 133, 6716929)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (16, 21, 6049)-net over F4, using
- net defined by OOA [i] based on linear OOA(421, 6049, F4, 5, 5) (dual of [(6049, 5), 30224, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(421, 12099, F4, 5) (dual of [12099, 12078, 6]-code), using
- trace code [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(421, 12099, F4, 5) (dual of [12099, 12078, 6]-code), using
- net defined by OOA [i] based on linear OOA(421, 6049, F4, 5, 5) (dual of [(6049, 5), 30224, 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 (16, 21, 6049)-net over F4, using
(123, 123+10, large)-Net over F4 — Digital
Digital (123, 133, large)-net over F4, using
- t-expansion [i] based on digital (121, 133, large)-net over F4, using
- 2 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
- 2 times m-reduction [i] based on digital (121, 135, large)-net over F4, using
(123, 123+10, large)-Net in Base 4 — Upper bound on s
There is no (123, 133, large)-net in base 4, because
- 8 times m-reduction [i] would yield (123, 125, large)-net in base 4, but