Best Known (219, 219+34, s)-Nets in Base 4
(219, 219+34, 61681)-Net over F4 — Constructive and digital
Digital (219, 253, 61681)-net over F4, using
- 42 times duplication [i] based on digital (217, 251, 61681)-net over F4, using
- net defined by OOA [i] based on linear OOA(4251, 61681, F4, 34, 34) (dual of [(61681, 34), 2096903, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(4251, 1048577, F4, 34) (dual of [1048577, 1048326, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4251, 1048586, F4, 34) (dual of [1048586, 1048335, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- linear OA(4251, 1048576, F4, 34) (dual of [1048576, 1048325, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4241, 1048576, F4, 33) (dual of [1048576, 1048335, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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(33) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(4251, 1048586, F4, 34) (dual of [1048586, 1048335, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(4251, 1048577, F4, 34) (dual of [1048577, 1048326, 35]-code), using
- net defined by OOA [i] based on linear OOA(4251, 61681, F4, 34, 34) (dual of [(61681, 34), 2096903, 35]-NRT-code), using
(219, 219+34, 349529)-Net over F4 — Digital
Digital (219, 253, 349529)-net over F4, using
- 41 times duplication [i] based on digital (218, 252, 349529)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4252, 349529, F4, 3, 34) (dual of [(349529, 3), 1048335, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4252, 1048587, F4, 34) (dual of [1048587, 1048335, 35]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4251, 1048586, F4, 34) (dual of [1048586, 1048335, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(32) [i] based on
- linear OA(4251, 1048576, F4, 34) (dual of [1048576, 1048325, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4241, 1048576, F4, 33) (dual of [1048576, 1048335, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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(33) ⊂ Ce(32) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4251, 1048586, F4, 34) (dual of [1048586, 1048335, 35]-code), using
- OOA 3-folding [i] based on linear OA(4252, 1048587, F4, 34) (dual of [1048587, 1048335, 35]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4252, 349529, F4, 3, 34) (dual of [(349529, 3), 1048335, 35]-NRT-code), using
(219, 219+34, large)-Net in Base 4 — Upper bound on s
There is no (219, 253, large)-net in base 4, because
- 32 times m-reduction [i] would yield (219, 221, large)-net in base 4, but