Best Known (163−22, 163, s)-Nets in Base 4
(163−22, 163, 95326)-Net over F4 — Constructive and digital
Digital (141, 163, 95326)-net over F4, using
- 42 times duplication [i] based on digital (139, 161, 95326)-net over F4, using
- net defined by OOA [i] based on linear OOA(4161, 95326, F4, 22, 22) (dual of [(95326, 22), 2097011, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4161, 1048586, F4, 22) (dual of [1048586, 1048425, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- OA 11-folding and stacking [i] based on linear OA(4161, 1048586, F4, 22) (dual of [1048586, 1048425, 23]-code), using
- net defined by OOA [i] based on linear OOA(4161, 95326, F4, 22, 22) (dual of [(95326, 22), 2097011, 23]-NRT-code), using
(163−22, 163, 349529)-Net over F4 — Digital
Digital (141, 163, 349529)-net over F4, using
- 41 times duplication [i] based on digital (140, 162, 349529)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4162, 349529, F4, 3, 22) (dual of [(349529, 3), 1048425, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4162, 1048587, F4, 22) (dual of [1048587, 1048425, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4161, 1048586, F4, 22) (dual of [1048586, 1048425, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4161, 1048576, F4, 22) (dual of [1048576, 1048415, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4161, 1048586, F4, 22) (dual of [1048586, 1048425, 23]-code), using
- OOA 3-folding [i] based on linear OA(4162, 1048587, F4, 22) (dual of [1048587, 1048425, 23]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4162, 349529, F4, 3, 22) (dual of [(349529, 3), 1048425, 23]-NRT-code), using
(163−22, 163, large)-Net in Base 4 — Upper bound on s
There is no (141, 163, large)-net in base 4, because
- 20 times m-reduction [i] would yield (141, 143, large)-net in base 4, but