Best Known (127, 127+10, s)-Nets in Base 4
(127, 127+10, 6743647)-Net over F4 — Constructive and digital
Digital (127, 137, 6743647)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (20, 25, 32767)-net over F4, using
- net defined by OOA [i] based on linear OOA(425, 32767, F4, 5, 5) (dual of [(32767, 5), 163810, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(425, 65535, F4, 5) (dual of [65535, 65510, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(425, 65535, F4, 5) (dual of [65535, 65510, 6]-code), using
- net defined by OOA [i] based on linear OOA(425, 32767, F4, 5, 5) (dual of [(32767, 5), 163810, 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 (20, 25, 32767)-net over F4, using
(127, 127+10, large)-Net over F4 — Digital
Digital (127, 137, large)-net over F4, using
- 42 times duplication [i] based on digital (125, 135, large)-net over F4, using
- t-expansion [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
- t-expansion [i] based on digital (121, 135, large)-net over F4, using
(127, 127+10, large)-Net in Base 4 — Upper bound on s
There is no (127, 137, large)-net in base 4, because
- 8 times m-reduction [i] would yield (127, 129, large)-net in base 4, but