Best Known (165−22, 165, s)-Nets in Base 4
(165−22, 165, 95327)-Net over F4 — Constructive and digital
Digital (143, 165, 95327)-net over F4, using
- 41 times duplication [i] based on digital (142, 164, 95327)-net over F4, using
- net defined by OOA [i] based on linear OOA(4164, 95327, F4, 22, 22) (dual of [(95327, 22), 2097030, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4164, 1048597, F4, 22) (dual of [1048597, 1048433, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [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(4141, 1048576, F4, 19) (dual of [1048576, 1048435, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- OA 11-folding and stacking [i] based on linear OA(4164, 1048597, F4, 22) (dual of [1048597, 1048433, 23]-code), using
- net defined by OOA [i] based on linear OOA(4164, 95327, F4, 22, 22) (dual of [(95327, 22), 2097030, 23]-NRT-code), using
(165−22, 165, 386472)-Net over F4 — Digital
Digital (143, 165, 386472)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4165, 386472, F4, 2, 22) (dual of [(386472, 2), 772779, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4165, 524300, F4, 2, 22) (dual of [(524300, 2), 1048435, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4165, 1048600, F4, 22) (dual of [1048600, 1048435, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [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(4141, 1048576, F4, 19) (dual of [1048576, 1048435, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(44, 24, F4, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- Hamming code H(4,4) [i]
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(4165, 1048600, F4, 22) (dual of [1048600, 1048435, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(4165, 524300, F4, 2, 22) (dual of [(524300, 2), 1048435, 23]-NRT-code), using
(165−22, 165, large)-Net in Base 4 — Upper bound on s
There is no (143, 165, large)-net in base 4, because
- 20 times m-reduction [i] would yield (143, 145, large)-net in base 4, but