Best Known (111, 135, s)-Nets in Base 2
(111, 135, 320)-Net over F2 — Constructive and digital
Digital (111, 135, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 27, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
(111, 135, 687)-Net over F2 — Digital
Digital (111, 135, 687)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2135, 687, F2, 3, 24) (dual of [(687, 3), 1926, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2135, 2061, F2, 24) (dual of [2061, 1926, 25]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2133, 2059, F2, 24) (dual of [2059, 1926, 25]-code), using
- 1 times truncation [i] based on linear OA(2134, 2060, F2, 25) (dual of [2060, 1926, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2133, 2048, F2, 25) (dual of [2048, 1915, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2122, 2048, F2, 23) (dual of [2048, 1926, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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 X applied to Ce(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(2134, 2060, F2, 25) (dual of [2060, 1926, 26]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2133, 2059, F2, 24) (dual of [2059, 1926, 25]-code), using
- OOA 3-folding [i] based on linear OA(2135, 2061, F2, 24) (dual of [2061, 1926, 25]-code), using
(111, 135, 12863)-Net in Base 2 — Upper bound on s
There is no (111, 135, 12864)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 43576 862727 045512 969705 680825 057656 523737 > 2135 [i]