Best Known (140, 140+25, s)-Nets in Base 4
(140, 140+25, 21846)-Net over F4 — Constructive and digital
Digital (140, 165, 21846)-net over F4, using
- 41 times duplication [i] based on digital (139, 164, 21846)-net over F4, using
- net defined by OOA [i] based on linear OOA(4164, 21846, F4, 25, 25) (dual of [(21846, 25), 545986, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4164, 262153, F4, 25) (dual of [262153, 261989, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4164, 262154, F4, 25) (dual of [262154, 261990, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 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(4154, 262144, F4, 23) (dual of [262144, 261990, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(41, 10, F4, 1) (dual of [10, 9, 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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4164, 262154, F4, 25) (dual of [262154, 261990, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4164, 262153, F4, 25) (dual of [262153, 261989, 26]-code), using
- net defined by OOA [i] based on linear OOA(4164, 21846, F4, 25, 25) (dual of [(21846, 25), 545986, 26]-NRT-code), using
(140, 140+25, 87385)-Net over F4 — Digital
Digital (140, 165, 87385)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4165, 87385, F4, 3, 25) (dual of [(87385, 3), 261990, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4165, 262155, F4, 25) (dual of [262155, 261990, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4164, 262154, F4, 25) (dual of [262154, 261990, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 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(4154, 262144, F4, 23) (dual of [262144, 261990, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(41, 10, F4, 1) (dual of [10, 9, 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(24) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4164, 262154, F4, 25) (dual of [262154, 261990, 26]-code), using
- OOA 3-folding [i] based on linear OA(4165, 262155, F4, 25) (dual of [262155, 261990, 26]-code), using
(140, 140+25, large)-Net in Base 4 — Upper bound on s
There is no (140, 165, large)-net in base 4, because
- 23 times m-reduction [i] would yield (140, 142, large)-net in base 4, but