Best Known (153−21, 153, s)-Nets in Base 4
(153−21, 153, 104858)-Net over F4 — Constructive and digital
Digital (132, 153, 104858)-net over F4, using
- 41 times duplication [i] based on digital (131, 152, 104858)-net over F4, using
- net defined by OOA [i] based on linear OOA(4152, 104858, F4, 21, 21) (dual of [(104858, 21), 2201866, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4152, 1048581, F4, 21) (dual of [1048581, 1048429, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4152, 1048587, F4, 21) (dual of [1048587, 1048435, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4141, 1048576, F4, 19) (dual of [1048576, 1048435, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(4152, 1048587, F4, 21) (dual of [1048587, 1048435, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4152, 1048581, F4, 21) (dual of [1048581, 1048429, 22]-code), using
- net defined by OOA [i] based on linear OOA(4152, 104858, F4, 21, 21) (dual of [(104858, 21), 2201866, 22]-NRT-code), using
(153−21, 153, 349529)-Net over F4 — Digital
Digital (132, 153, 349529)-net over F4, using
- 41 times duplication [i] based on digital (131, 152, 349529)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4152, 349529, F4, 3, 21) (dual of [(349529, 3), 1048435, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4152, 1048587, F4, 21) (dual of [1048587, 1048435, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4141, 1048576, F4, 19) (dual of [1048576, 1048435, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 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(20) ⊂ Ce(18) [i] based on
- OOA 3-folding [i] based on linear OA(4152, 1048587, F4, 21) (dual of [1048587, 1048435, 22]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4152, 349529, F4, 3, 21) (dual of [(349529, 3), 1048435, 22]-NRT-code), using
(153−21, 153, large)-Net in Base 4 — Upper bound on s
There is no (132, 153, large)-net in base 4, because
- 19 times m-reduction [i] would yield (132, 134, large)-net in base 4, but