Best Known (119, 119+18, s)-Nets in Base 4
(119, 119+18, 116512)-Net over F4 — Constructive and digital
Digital (119, 137, 116512)-net over F4, using
- 41 times duplication [i] based on digital (118, 136, 116512)-net over F4, using
- net defined by OOA [i] based on linear OOA(4136, 116512, F4, 18, 18) (dual of [(116512, 18), 2097080, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4136, 1048608, F4, 18) (dual of [1048608, 1048472, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4136, 1048611, F4, 18) (dual of [1048611, 1048475, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4101, 1048576, F4, 14) (dual of [1048576, 1048475, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4136, 1048611, F4, 18) (dual of [1048611, 1048475, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(4136, 1048608, F4, 18) (dual of [1048608, 1048472, 19]-code), using
- net defined by OOA [i] based on linear OOA(4136, 116512, F4, 18, 18) (dual of [(116512, 18), 2097080, 19]-NRT-code), using
(119, 119+18, 524306)-Net over F4 — Digital
Digital (119, 137, 524306)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4137, 524306, F4, 2, 18) (dual of [(524306, 2), 1048475, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4137, 1048612, F4, 18) (dual of [1048612, 1048475, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4136, 1048611, F4, 18) (dual of [1048611, 1048475, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(4101, 1048576, F4, 14) (dual of [1048576, 1048475, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4136, 1048611, F4, 18) (dual of [1048611, 1048475, 19]-code), using
- OOA 2-folding [i] based on linear OA(4137, 1048612, F4, 18) (dual of [1048612, 1048475, 19]-code), using
(119, 119+18, large)-Net in Base 4 — Upper bound on s
There is no (119, 137, large)-net in base 4, because
- 16 times m-reduction [i] would yield (119, 121, large)-net in base 4, but