Best Known (167, 167+28, s)-Nets in Base 4
(167, 167+28, 18726)-Net over F4 — Constructive and digital
Digital (167, 195, 18726)-net over F4, using
- 42 times duplication [i] based on digital (165, 193, 18726)-net over F4, using
- t-expansion [i] based on digital (164, 193, 18726)-net over F4, using
- net defined by OOA [i] based on linear OOA(4193, 18726, F4, 29, 29) (dual of [(18726, 29), 542861, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4193, 262165, F4, 29) (dual of [262165, 261972, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(28) ⊂ Ce(25) [i] based on
- OOA 14-folding and stacking with additional row [i] based on linear OA(4193, 262165, F4, 29) (dual of [262165, 261972, 30]-code), using
- net defined by OOA [i] based on linear OOA(4193, 18726, F4, 29, 29) (dual of [(18726, 29), 542861, 30]-NRT-code), using
- t-expansion [i] based on digital (164, 193, 18726)-net over F4, using
(167, 167+28, 131088)-Net over F4 — Digital
Digital (167, 195, 131088)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4195, 131088, F4, 2, 28) (dual of [(131088, 2), 261981, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4195, 262176, F4, 28) (dual of [262176, 261981, 29]-code), using
- strength reduction [i] based on linear OA(4195, 262176, F4, 29) (dual of [262176, 261981, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- strength reduction [i] based on linear OA(4195, 262176, F4, 29) (dual of [262176, 261981, 30]-code), using
- OOA 2-folding [i] based on linear OA(4195, 262176, F4, 28) (dual of [262176, 261981, 29]-code), using
(167, 167+28, large)-Net in Base 4 — Upper bound on s
There is no (167, 195, large)-net in base 4, because
- 26 times m-reduction [i] would yield (167, 169, large)-net in base 4, but