Best Known (167, 191, s)-Nets in Base 4
(167, 191, 87386)-Net over F4 — Constructive and digital
Digital (167, 191, 87386)-net over F4, using
- net defined by OOA [i] based on linear OOA(4191, 87386, F4, 24, 24) (dual of [(87386, 24), 2097073, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(4191, 1048632, F4, 24) (dual of [1048632, 1048441, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 1048636, F4, 24) (dual of [1048636, 1048445, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4191, 1048636, F4, 24) (dual of [1048636, 1048445, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(4191, 1048632, F4, 24) (dual of [1048632, 1048441, 25]-code), using
(167, 191, 524318)-Net over F4 — Digital
Digital (167, 191, 524318)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4191, 524318, F4, 2, 24) (dual of [(524318, 2), 1048445, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4191, 1048636, F4, 24) (dual of [1048636, 1048445, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(4191, 1048636, F4, 24) (dual of [1048636, 1048445, 25]-code), using
(167, 191, large)-Net in Base 4 — Upper bound on s
There is no (167, 191, large)-net in base 4, because
- 22 times m-reduction [i] would yield (167, 169, large)-net in base 4, but