Best Known (123, 123+20, s)-Nets in Base 4
(123, 123+20, 26218)-Net over F4 — Constructive and digital
Digital (123, 143, 26218)-net over F4, using
- net defined by OOA [i] based on linear OOA(4143, 26218, F4, 20, 20) (dual of [(26218, 20), 524217, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4143, 262180, F4, 20) (dual of [262180, 262037, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4143, 262187, F4, 20) (dual of [262187, 262044, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4100, 262144, F4, 15) (dual of [262144, 262044, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4143, 262187, F4, 20) (dual of [262187, 262044, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4143, 262180, F4, 20) (dual of [262180, 262037, 21]-code), using
(123, 123+20, 141433)-Net over F4 — Digital
Digital (123, 143, 141433)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4143, 141433, F4, 20) (dual of [141433, 141290, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4143, 262187, F4, 20) (dual of [262187, 262044, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4100, 262144, F4, 15) (dual of [262144, 262044, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4143, 262187, F4, 20) (dual of [262187, 262044, 21]-code), using
(123, 123+20, large)-Net in Base 4 — Upper bound on s
There is no (123, 143, large)-net in base 4, because
- 18 times m-reduction [i] would yield (123, 125, large)-net in base 4, but