Best Known (252−30, 252, s)-Nets in Base 4
(252−30, 252, 279623)-Net over F4 — Constructive and digital
Digital (222, 252, 279623)-net over F4, using
- 42 times duplication [i] based on digital (220, 250, 279623)-net over F4, using
- net defined by OOA [i] based on linear OOA(4250, 279623, F4, 30, 30) (dual of [(279623, 30), 8388440, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4250, 4194345, F4, 30) (dual of [4194345, 4194095, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4250, 4194347, F4, 30) (dual of [4194347, 4194097, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- linear OA(4243, 4194304, F4, 30) (dual of [4194304, 4194061, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4199, 4194304, F4, 25) (dual of [4194304, 4194105, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4250, 4194347, F4, 30) (dual of [4194347, 4194097, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(4250, 4194345, F4, 30) (dual of [4194345, 4194095, 31]-code), using
- net defined by OOA [i] based on linear OOA(4250, 279623, F4, 30, 30) (dual of [(279623, 30), 8388440, 31]-NRT-code), using
(252−30, 252, 1398119)-Net over F4 — Digital
Digital (222, 252, 1398119)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4252, 1398119, F4, 3, 30) (dual of [(1398119, 3), 4194105, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4252, 4194357, F4, 30) (dual of [4194357, 4194105, 31]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4251, 4194356, F4, 30) (dual of [4194356, 4194105, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- linear OA(4243, 4194304, F4, 30) (dual of [4194304, 4194061, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4199, 4194304, F4, 25) (dual of [4194304, 4194105, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(48, 52, F4, 4) (dual of [52, 44, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(48, 85, F4, 4) (dual of [85, 77, 5]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4251, 4194356, F4, 30) (dual of [4194356, 4194105, 31]-code), using
- OOA 3-folding [i] based on linear OA(4252, 4194357, F4, 30) (dual of [4194357, 4194105, 31]-code), using
(252−30, 252, large)-Net in Base 4 — Upper bound on s
There is no (222, 252, large)-net in base 4, because
- 28 times m-reduction [i] would yield (222, 224, large)-net in base 4, but