Best Known (242−29, 242, s)-Nets in Base 4
(242−29, 242, 299596)-Net over F4 — Constructive and digital
Digital (213, 242, 299596)-net over F4, using
- 42 times duplication [i] based on digital (211, 240, 299596)-net over F4, using
- net defined by OOA [i] based on linear OOA(4240, 299596, F4, 29, 29) (dual of [(299596, 29), 8688044, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4240, 4194345, F4, 29) (dual of [4194345, 4194105, 30]-code), using
- 3 times code embedding in larger space [i] based on linear OA(4237, 4194342, F4, 29) (dual of [4194342, 4194105, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(45, 38, F4, 3) (dual of [38, 33, 4]-code or 38-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
- 3 times code embedding in larger space [i] based on linear OA(4237, 4194342, F4, 29) (dual of [4194342, 4194105, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4240, 4194345, F4, 29) (dual of [4194345, 4194105, 30]-code), using
- net defined by OOA [i] based on linear OOA(4240, 299596, F4, 29, 29) (dual of [(299596, 29), 8688044, 30]-NRT-code), using
(242−29, 242, 1398119)-Net over F4 — Digital
Digital (213, 242, 1398119)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4242, 1398119, F4, 3, 29) (dual of [(1398119, 3), 4194115, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4242, 4194357, F4, 29) (dual of [4194357, 4194115, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4242, 4194358, F4, 29) (dual of [4194358, 4194116, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(410, 54, F4, 5) (dual of [54, 44, 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(28) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4242, 4194358, F4, 29) (dual of [4194358, 4194116, 30]-code), using
- OOA 3-folding [i] based on linear OA(4242, 4194357, F4, 29) (dual of [4194357, 4194115, 30]-code), using
(242−29, 242, large)-Net in Base 4 — Upper bound on s
There is no (213, 242, large)-net in base 4, because
- 27 times m-reduction [i] would yield (213, 215, large)-net in base 4, but