Best Known (211, 211+35, s)-Nets in Base 4
(211, 211+35, 15423)-Net over F4 — Constructive and digital
Digital (211, 246, 15423)-net over F4, using
- 41 times duplication [i] based on digital (210, 245, 15423)-net over F4, using
- net defined by OOA [i] based on linear OOA(4245, 15423, F4, 35, 35) (dual of [(15423, 35), 539560, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4245, 262192, F4, 35) (dual of [262192, 261947, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4245, 262199, F4, 35) (dual of [262199, 261954, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(4235, 262144, F4, 35) (dual of [262144, 261909, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(410, 55, F4, 5) (dual of [55, 45, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4245, 262199, F4, 35) (dual of [262199, 261954, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4245, 262192, F4, 35) (dual of [262192, 261947, 36]-code), using
- net defined by OOA [i] based on linear OOA(4245, 15423, F4, 35, 35) (dual of [(15423, 35), 539560, 36]-NRT-code), using
(211, 211+35, 131100)-Net over F4 — Digital
Digital (211, 246, 131100)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4246, 131100, F4, 2, 35) (dual of [(131100, 2), 261954, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4246, 262200, F4, 35) (dual of [262200, 261954, 36]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4245, 262199, F4, 35) (dual of [262199, 261954, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(4235, 262144, F4, 35) (dual of [262144, 261909, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(410, 55, F4, 5) (dual of [55, 45, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4245, 262199, F4, 35) (dual of [262199, 261954, 36]-code), using
- OOA 2-folding [i] based on linear OA(4246, 262200, F4, 35) (dual of [262200, 261954, 36]-code), using
(211, 211+35, large)-Net in Base 4 — Upper bound on s
There is no (211, 246, large)-net in base 4, because
- 33 times m-reduction [i] would yield (211, 213, large)-net in base 4, but