Best Known (186−31, 186, s)-Nets in Base 4
(186−31, 186, 4370)-Net over F4 — Constructive and digital
Digital (155, 186, 4370)-net over F4, using
- net defined by OOA [i] based on linear OOA(4186, 4370, F4, 31, 31) (dual of [(4370, 31), 135284, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4186, 65551, F4, 31) (dual of [65551, 65365, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4186, 65553, F4, 31) (dual of [65553, 65367, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4186, 65553, F4, 31) (dual of [65553, 65367, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4186, 65551, F4, 31) (dual of [65551, 65365, 32]-code), using
(186−31, 186, 32776)-Net over F4 — Digital
Digital (155, 186, 32776)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4186, 32776, F4, 2, 31) (dual of [(32776, 2), 65366, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4186, 65552, F4, 31) (dual of [65552, 65366, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4186, 65553, F4, 31) (dual of [65553, 65367, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4186, 65553, F4, 31) (dual of [65553, 65367, 32]-code), using
- OOA 2-folding [i] based on linear OA(4186, 65552, F4, 31) (dual of [65552, 65366, 32]-code), using
(186−31, 186, large)-Net in Base 4 — Upper bound on s
There is no (155, 186, large)-net in base 4, because
- 29 times m-reduction [i] would yield (155, 157, large)-net in base 4, but