Best Known (206−18, 206, s)-Nets in Base 2
(206−18, 206, 466039)-Net over F2 — Constructive and digital
Digital (188, 206, 466039)-net over F2, using
- t-expansion [i] based on digital (187, 206, 466039)-net over F2, using
- net defined by OOA [i] based on linear OOA(2206, 466039, F2, 19, 19) (dual of [(466039, 19), 8854535, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2206, 4194352, F2, 19) (dual of [4194352, 4194146, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2206, 4194355, F2, 19) (dual of [4194355, 4194149, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2199, 4194304, F2, 19) (dual of [4194304, 4194105, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2155, 4194304, F2, 15) (dual of [4194304, 4194149, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(27, 51, F2, 3) (dual of [51, 44, 4]-code or 51-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2206, 4194355, F2, 19) (dual of [4194355, 4194149, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2206, 4194352, F2, 19) (dual of [4194352, 4194146, 20]-code), using
- net defined by OOA [i] based on linear OOA(2206, 466039, F2, 19, 19) (dual of [(466039, 19), 8854535, 20]-NRT-code), using
(206−18, 206, 699059)-Net over F2 — Digital
Digital (188, 206, 699059)-net over F2, using
- 21 times duplication [i] based on digital (187, 205, 699059)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2205, 699059, F2, 6, 18) (dual of [(699059, 6), 4194149, 19]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2205, 4194354, F2, 18) (dual of [4194354, 4194149, 19]-code), using
- 1 times truncation [i] based on linear OA(2206, 4194355, F2, 19) (dual of [4194355, 4194149, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2199, 4194304, F2, 19) (dual of [4194304, 4194105, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2155, 4194304, F2, 15) (dual of [4194304, 4194149, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(27, 51, F2, 3) (dual of [51, 44, 4]-code or 51-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2206, 4194355, F2, 19) (dual of [4194355, 4194149, 20]-code), using
- OOA 6-folding [i] based on linear OA(2205, 4194354, F2, 18) (dual of [4194354, 4194149, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2205, 699059, F2, 6, 18) (dual of [(699059, 6), 4194149, 19]-NRT-code), using
(206−18, 206, large)-Net in Base 2 — Upper bound on s
There is no (188, 206, large)-net in base 2, because
- 16 times m-reduction [i] would yield (188, 190, large)-net in base 2, but