Best Known (124, 136, s)-Nets in Base 2
(124, 136, 699055)-Net over F2 — Constructive and digital
Digital (124, 136, 699055)-net over F2, using
- net defined by OOA [i] based on linear OOA(2136, 699055, F2, 12, 12) (dual of [(699055, 12), 8388524, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2136, 4194330, F2, 12) (dual of [4194330, 4194194, 13]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2133, 4194327, F2, 12) (dual of [4194327, 4194194, 13]-code), using
- 1 times truncation [i] based on linear OA(2134, 4194328, F2, 13) (dual of [4194328, 4194194, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2133, 4194304, F2, 13) (dual of [4194304, 4194171, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2111, 4194304, F2, 11) (dual of [4194304, 4194193, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(223, 24, F2, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
- dual of repetition code with length 24 [i]
- linear OA(21, 24, F2, 1) (dual of [24, 23, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(2134, 4194328, F2, 13) (dual of [4194328, 4194194, 14]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2133, 4194327, F2, 12) (dual of [4194327, 4194194, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2136, 4194330, F2, 12) (dual of [4194330, 4194194, 13]-code), using
(124, 136, 1048582)-Net over F2 — Digital
Digital (124, 136, 1048582)-net over F2, using
- 22 times duplication [i] based on digital (122, 134, 1048582)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2134, 1048582, F2, 4, 12) (dual of [(1048582, 4), 4194194, 13]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2134, 4194328, F2, 12) (dual of [4194328, 4194194, 13]-code), using
- strength reduction [i] based on linear OA(2134, 4194328, F2, 13) (dual of [4194328, 4194194, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2133, 4194304, F2, 13) (dual of [4194304, 4194171, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2111, 4194304, F2, 11) (dual of [4194304, 4194193, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(223, 24, F2, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
- dual of repetition code with length 24 [i]
- linear OA(21, 24, F2, 1) (dual of [24, 23, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- strength reduction [i] based on linear OA(2134, 4194328, F2, 13) (dual of [4194328, 4194194, 14]-code), using
- OOA 4-folding [i] based on linear OA(2134, 4194328, F2, 12) (dual of [4194328, 4194194, 13]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2134, 1048582, F2, 4, 12) (dual of [(1048582, 4), 4194194, 13]-NRT-code), using
(124, 136, large)-Net in Base 2 — Upper bound on s
There is no (124, 136, large)-net in base 2, because
- 10 times m-reduction [i] would yield (124, 126, large)-net in base 2, but