Best Known (119−18, 119, s)-Nets in Base 4
(119−18, 119, 29128)-Net over F4 — Constructive and digital
Digital (101, 119, 29128)-net over F4, using
- 41 times duplication [i] based on digital (100, 118, 29128)-net over F4, using
- net defined by OOA [i] based on linear OOA(4118, 29128, F4, 18, 18) (dual of [(29128, 18), 524186, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4118, 262152, F4, 18) (dual of [262152, 262034, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4118, 262153, F4, 18) (dual of [262153, 262035, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [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(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4118, 262153, F4, 18) (dual of [262153, 262035, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(4118, 262152, F4, 18) (dual of [262152, 262034, 19]-code), using
- net defined by OOA [i] based on linear OOA(4118, 29128, F4, 18, 18) (dual of [(29128, 18), 524186, 19]-NRT-code), using
(119−18, 119, 106333)-Net over F4 — Digital
Digital (101, 119, 106333)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4119, 106333, F4, 2, 18) (dual of [(106333, 2), 212547, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4119, 131077, F4, 2, 18) (dual of [(131077, 2), 262035, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4119, 262154, F4, 18) (dual of [262154, 262035, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4118, 262153, F4, 18) (dual of [262153, 262035, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [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(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4118, 262153, F4, 18) (dual of [262153, 262035, 19]-code), using
- OOA 2-folding [i] based on linear OA(4119, 262154, F4, 18) (dual of [262154, 262035, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(4119, 131077, F4, 2, 18) (dual of [(131077, 2), 262035, 19]-NRT-code), using
(119−18, 119, large)-Net in Base 4 — Upper bound on s
There is no (101, 119, large)-net in base 4, because
- 16 times m-reduction [i] would yield (101, 103, large)-net in base 4, but