Best Known (217−29, 217, s)-Nets in Base 4
(217−29, 217, 74900)-Net over F4 — Constructive and digital
Digital (188, 217, 74900)-net over F4, using
- 41 times duplication [i] based on digital (187, 216, 74900)-net over F4, using
- net defined by OOA [i] based on linear OOA(4216, 74900, F4, 29, 29) (dual of [(74900, 29), 2171884, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4216, 1048601, F4, 29) (dual of [1048601, 1048385, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4216, 1048611, F4, 29) (dual of [1048611, 1048395, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(28) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4216, 1048611, F4, 29) (dual of [1048611, 1048395, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4216, 1048601, F4, 29) (dual of [1048601, 1048385, 30]-code), using
- net defined by OOA [i] based on linear OOA(4216, 74900, F4, 29, 29) (dual of [(74900, 29), 2171884, 30]-NRT-code), using
(217−29, 217, 349537)-Net over F4 — Digital
Digital (188, 217, 349537)-net over F4, using
- 41 times duplication [i] based on digital (187, 216, 349537)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4216, 349537, F4, 3, 29) (dual of [(349537, 3), 1048395, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4216, 1048611, F4, 29) (dual of [1048611, 1048395, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(28) ⊂ Ce(24) [i] based on
- OOA 3-folding [i] based on linear OA(4216, 1048611, F4, 29) (dual of [1048611, 1048395, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4216, 349537, F4, 3, 29) (dual of [(349537, 3), 1048395, 30]-NRT-code), using
(217−29, 217, large)-Net in Base 4 — Upper bound on s
There is no (188, 217, large)-net in base 4, because
- 27 times m-reduction [i] would yield (188, 190, large)-net in base 4, but