Best Known (141, 141+11, s)-Nets in Base 4
(141, 141+11, 6841951)-Net over F4 — Constructive and digital
Digital (141, 152, 6841951)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (23, 28, 131071)-net over F4, using
- net defined by OOA [i] based on linear OOA(428, 131071, F4, 5, 5) (dual of [(131071, 5), 655327, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(428, 262143, F4, 5) (dual of [262143, 262115, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(428, 262143, F4, 5) (dual of [262143, 262115, 6]-code), using
- net defined by OOA [i] based on linear OOA(428, 131071, F4, 5, 5) (dual of [(131071, 5), 655327, 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 (23, 28, 131071)-net over F4, using
(141, 141+11, large)-Net over F4 — Digital
Digital (141, 152, large)-net over F4, using
- t-expansion [i] based on digital (139, 152, large)-net over F4, using
- 3 times m-reduction [i] based on digital (139, 155, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4155, large, F4, 16) (dual of [large, large−155, 17]-code), using
- 11 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 11 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4155, large, F4, 16) (dual of [large, large−155, 17]-code), using
- 3 times m-reduction [i] based on digital (139, 155, large)-net over F4, using
(141, 141+11, large)-Net in Base 4 — Upper bound on s
There is no (141, 152, large)-net in base 4, because
- 9 times m-reduction [i] would yield (141, 143, large)-net in base 4, but