Best Known (235−30, 235, s)-Nets in Base 2
(235−30, 235, 2187)-Net over F2 — Constructive and digital
Digital (205, 235, 2187)-net over F2, using
- 21 times duplication [i] based on digital (204, 234, 2187)-net over F2, using
- t-expansion [i] based on digital (203, 234, 2187)-net over F2, using
- net defined by OOA [i] based on linear OOA(2234, 2187, F2, 31, 31) (dual of [(2187, 31), 67563, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2234, 32806, F2, 31) (dual of [32806, 32572, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2233, 32805, F2, 31) (dual of [32805, 32572, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2196, 32768, F2, 27) (dual of [32768, 32572, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(27, 37, F2, 3) (dual of [37, 30, 4]-code or 37-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(30) ⊂ Ce(26) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2233, 32805, F2, 31) (dual of [32805, 32572, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2234, 32806, F2, 31) (dual of [32806, 32572, 32]-code), using
- net defined by OOA [i] based on linear OOA(2234, 2187, F2, 31, 31) (dual of [(2187, 31), 67563, 32]-NRT-code), using
- t-expansion [i] based on digital (203, 234, 2187)-net over F2, using
(235−30, 235, 6561)-Net over F2 — Digital
Digital (205, 235, 6561)-net over F2, using
- 22 times duplication [i] based on digital (203, 233, 6561)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2233, 6561, F2, 5, 30) (dual of [(6561, 5), 32572, 31]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2233, 32805, F2, 30) (dual of [32805, 32572, 31]-code), using
- strength reduction [i] based on linear OA(2233, 32805, F2, 31) (dual of [32805, 32572, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2196, 32768, F2, 27) (dual of [32768, 32572, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(27, 37, F2, 3) (dual of [37, 30, 4]-code or 37-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(30) ⊂ Ce(26) [i] based on
- strength reduction [i] based on linear OA(2233, 32805, F2, 31) (dual of [32805, 32572, 32]-code), using
- OOA 5-folding [i] based on linear OA(2233, 32805, F2, 30) (dual of [32805, 32572, 31]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2233, 6561, F2, 5, 30) (dual of [(6561, 5), 32572, 31]-NRT-code), using
(235−30, 235, 334098)-Net in Base 2 — Upper bound on s
There is no (205, 235, 334099)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 55214 985734 207225 177268 772562 598376 199147 866561 242532 727025 022658 380616 > 2235 [i]