Best Known (138, 138+22, s)-Nets in Base 2
(138, 138+22, 1492)-Net over F2 — Constructive and digital
Digital (138, 160, 1492)-net over F2, using
- net defined by OOA [i] based on linear OOA(2160, 1492, F2, 22, 22) (dual of [(1492, 22), 32664, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(2160, 16412, F2, 22) (dual of [16412, 16252, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2160, 16415, F2, 22) (dual of [16415, 16255, 23]-code), using
- 1 times truncation [i] based on linear OA(2161, 16416, F2, 23) (dual of [16416, 16255, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2155, 16384, F2, 23) (dual of [16384, 16229, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- 1 times truncation [i] based on linear OA(2161, 16416, F2, 23) (dual of [16416, 16255, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2160, 16415, F2, 22) (dual of [16415, 16255, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(2160, 16412, F2, 22) (dual of [16412, 16252, 23]-code), using
(138, 138+22, 4103)-Net over F2 — Digital
Digital (138, 160, 4103)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2160, 4103, F2, 4, 22) (dual of [(4103, 4), 16252, 23]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2160, 16412, F2, 22) (dual of [16412, 16252, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2160, 16415, F2, 22) (dual of [16415, 16255, 23]-code), using
- 1 times truncation [i] based on linear OA(2161, 16416, F2, 23) (dual of [16416, 16255, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2155, 16384, F2, 23) (dual of [16384, 16229, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- 1 times truncation [i] based on linear OA(2161, 16416, F2, 23) (dual of [16416, 16255, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2160, 16415, F2, 22) (dual of [16415, 16255, 23]-code), using
- OOA 4-folding [i] based on linear OA(2160, 16412, F2, 22) (dual of [16412, 16252, 23]-code), using
(138, 138+22, 117374)-Net in Base 2 — Upper bound on s
There is no (138, 160, 117375)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1 461595 059048 488987 036387 815546 542039 457973 303126 > 2160 [i]