Best Known (133−12, 133, s)-Nets in Base 2
(133−12, 133, 699054)-Net over F2 — Constructive and digital
Digital (121, 133, 699054)-net over F2, using
- net defined by OOA [i] based on linear OOA(2133, 699054, F2, 12, 12) (dual of [(699054, 12), 8388515, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2133, 4194324, F2, 12) (dual of [4194324, 4194191, 13]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code 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(2133, 4194324, F2, 12) (dual of [4194324, 4194191, 13]-code), using
(133−12, 133, 1048581)-Net over F2 — Digital
Digital (121, 133, 1048581)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2133, 1048581, F2, 4, 12) (dual of [(1048581, 4), 4194191, 13]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2133, 4194324, F2, 12) (dual of [4194324, 4194191, 13]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2133, 4194327, F2, 12) (dual of [4194327, 4194194, 13]-code), using
- OOA 4-folding [i] based on linear OA(2133, 4194324, F2, 12) (dual of [4194324, 4194191, 13]-code), using
(133−12, 133, large)-Net in Base 2 — Upper bound on s
There is no (121, 133, large)-net in base 2, because
- 10 times m-reduction [i] would yield (121, 123, large)-net in base 2, but