Best Known (205, 205+20, s)-Nets in Base 2
(205, 205+20, 419433)-Net over F2 — Constructive and digital
Digital (205, 225, 419433)-net over F2, using
- t-expansion [i] based on digital (204, 225, 419433)-net over F2, using
- net defined by OOA [i] based on linear OOA(2225, 419433, F2, 21, 21) (dual of [(419433, 21), 8807868, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2225, 4194331, F2, 21) (dual of [4194331, 4194106, 22]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2222, 4194328, F2, 21) (dual of [4194328, 4194106, 22]-code), using
- construction X4 applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2221, 4194304, F2, 21) (dual of [4194304, 4194083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(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(20) ⊂ Ce(18) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2222, 4194328, F2, 21) (dual of [4194328, 4194106, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2225, 4194331, F2, 21) (dual of [4194331, 4194106, 22]-code), using
- net defined by OOA [i] based on linear OOA(2225, 419433, F2, 21, 21) (dual of [(419433, 21), 8807868, 22]-NRT-code), using
(205, 205+20, 667716)-Net over F2 — Digital
Digital (205, 225, 667716)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2225, 667716, F2, 6, 20) (dual of [(667716, 6), 4006071, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2225, 699055, F2, 6, 20) (dual of [(699055, 6), 4194105, 21]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2224, 699055, F2, 6, 20) (dual of [(699055, 6), 4194106, 21]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2224, 4194330, F2, 20) (dual of [4194330, 4194106, 21]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2221, 4194327, F2, 20) (dual of [4194327, 4194106, 21]-code), using
- 1 times truncation [i] based on linear OA(2222, 4194328, F2, 21) (dual of [4194328, 4194106, 22]-code), using
- construction X4 applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2221, 4194304, F2, 21) (dual of [4194304, 4194083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(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(20) ⊂ Ce(18) [i] based on
- 1 times truncation [i] based on linear OA(2222, 4194328, F2, 21) (dual of [4194328, 4194106, 22]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2221, 4194327, F2, 20) (dual of [4194327, 4194106, 21]-code), using
- OOA 6-folding [i] based on linear OA(2224, 4194330, F2, 20) (dual of [4194330, 4194106, 21]-code), using
- 21 times duplication [i] based on linear OOA(2224, 699055, F2, 6, 20) (dual of [(699055, 6), 4194106, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2225, 699055, F2, 6, 20) (dual of [(699055, 6), 4194105, 21]-NRT-code), using
(205, 205+20, large)-Net in Base 2 — Upper bound on s
There is no (205, 225, large)-net in base 2, because
- 18 times m-reduction [i] would yield (205, 207, large)-net in base 2, but