Best Known (218−31, 218, s)-Nets in Base 4
(218−31, 218, 17479)-Net over F4 — Constructive and digital
Digital (187, 218, 17479)-net over F4, using
- 43 times duplication [i] based on digital (184, 215, 17479)-net over F4, using
- net defined by OOA [i] based on linear OOA(4215, 17479, F4, 31, 31) (dual of [(17479, 31), 541634, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4215, 262186, F4, 31) (dual of [262186, 261971, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4215, 262187, F4, 31) (dual of [262187, 261972, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4215, 262187, F4, 31) (dual of [262187, 261972, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4215, 262186, F4, 31) (dual of [262186, 261971, 32]-code), using
- net defined by OOA [i] based on linear OOA(4215, 17479, F4, 31, 31) (dual of [(17479, 31), 541634, 32]-NRT-code), using
(218−31, 218, 131099)-Net over F4 — Digital
Digital (187, 218, 131099)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4218, 131099, F4, 2, 31) (dual of [(131099, 2), 261980, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4218, 262198, F4, 31) (dual of [262198, 261980, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4218, 262199, F4, 31) (dual of [262199, 261981, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(4208, 262144, F4, 31) (dual of [262144, 261936, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(410, 55, F4, 5) (dual of [55, 45, 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(30) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4218, 262199, F4, 31) (dual of [262199, 261981, 32]-code), using
- OOA 2-folding [i] based on linear OA(4218, 262198, F4, 31) (dual of [262198, 261980, 32]-code), using
(218−31, 218, large)-Net in Base 4 — Upper bound on s
There is no (187, 218, large)-net in base 4, because
- 29 times m-reduction [i] would yield (187, 189, large)-net in base 4, but