Best Known (222, 256, s)-Nets in Base 4
(222, 256, 61683)-Net over F4 — Constructive and digital
Digital (222, 256, 61683)-net over F4, using
- net defined by OOA [i] based on linear OOA(4256, 61683, F4, 34, 34) (dual of [(61683, 34), 2096966, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(4256, 1048611, F4, 34) (dual of [1048611, 1048355, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(29) [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(4221, 1048576, F4, 30) (dual of [1048576, 1048355, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(33) ⊂ Ce(29) [i] based on
- OA 17-folding and stacking [i] based on linear OA(4256, 1048611, F4, 34) (dual of [1048611, 1048355, 35]-code), using
(222, 256, 354723)-Net over F4 — Digital
Digital (222, 256, 354723)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4256, 354723, F4, 2, 34) (dual of [(354723, 2), 709190, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4256, 524305, F4, 2, 34) (dual of [(524305, 2), 1048354, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4256, 1048610, F4, 34) (dual of [1048610, 1048354, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4256, 1048611, F4, 34) (dual of [1048611, 1048355, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(29) [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(4221, 1048576, F4, 30) (dual of [1048576, 1048355, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(33) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4256, 1048611, F4, 34) (dual of [1048611, 1048355, 35]-code), using
- OOA 2-folding [i] based on linear OA(4256, 1048610, F4, 34) (dual of [1048610, 1048354, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(4256, 524305, F4, 2, 34) (dual of [(524305, 2), 1048354, 35]-NRT-code), using
(222, 256, large)-Net in Base 4 — Upper bound on s
There is no (222, 256, large)-net in base 4, because
- 32 times m-reduction [i] would yield (222, 224, large)-net in base 4, but