Best Known (104, 104+18, s)-Nets in Base 4
(104, 104+18, 29129)-Net over F4 — Constructive and digital
Digital (104, 122, 29129)-net over F4, using
- 41 times duplication [i] based on digital (103, 121, 29129)-net over F4, using
- net defined by OOA [i] based on linear OOA(4121, 29129, F4, 18, 18) (dual of [(29129, 18), 524201, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4121, 262161, F4, 18) (dual of [262161, 262040, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4121, 262165, F4, 18) (dual of [262165, 262044, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4121, 262165, F4, 18) (dual of [262165, 262044, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(4121, 262161, F4, 18) (dual of [262161, 262040, 19]-code), using
- net defined by OOA [i] based on linear OOA(4121, 29129, F4, 18, 18) (dual of [(29129, 18), 524201, 19]-NRT-code), using
(104, 104+18, 131083)-Net over F4 — Digital
Digital (104, 122, 131083)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4122, 131083, F4, 2, 18) (dual of [(131083, 2), 262044, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4122, 262166, F4, 18) (dual of [262166, 262044, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4121, 262165, F4, 18) (dual of [262165, 262044, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4121, 262165, F4, 18) (dual of [262165, 262044, 19]-code), using
- OOA 2-folding [i] based on linear OA(4122, 262166, F4, 18) (dual of [262166, 262044, 19]-code), using
(104, 104+18, large)-Net in Base 4 — Upper bound on s
There is no (104, 122, large)-net in base 4, because
- 16 times m-reduction [i] would yield (104, 106, large)-net in base 4, but