Best Known (209−29, 209, s)-Nets in Base 2
(209−29, 209, 1173)-Net over F2 — Constructive and digital
Digital (180, 209, 1173)-net over F2, using
- 22 times duplication [i] based on digital (178, 207, 1173)-net over F2, using
- net defined by OOA [i] based on linear OOA(2207, 1173, F2, 29, 29) (dual of [(1173, 29), 33810, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2207, 16423, F2, 29) (dual of [16423, 16216, 30]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2204, 16420, F2, 29) (dual of [16420, 16216, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2197, 16385, F2, 29) (dual of [16385, 16188, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2169, 16385, F2, 25) (dual of [16385, 16216, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-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 C([0,14]) ⊂ C([0,12]) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2204, 16420, F2, 29) (dual of [16420, 16216, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2207, 16423, F2, 29) (dual of [16423, 16216, 30]-code), using
- net defined by OOA [i] based on linear OOA(2207, 1173, F2, 29, 29) (dual of [(1173, 29), 33810, 30]-NRT-code), using
(209−29, 209, 3618)-Net over F2 — Digital
Digital (180, 209, 3618)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2209, 3618, F2, 4, 29) (dual of [(3618, 4), 14263, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2209, 4108, F2, 4, 29) (dual of [(4108, 4), 16223, 30]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2209, 16432, F2, 29) (dual of [16432, 16223, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(2197, 16384, F2, 29) (dual of [16384, 16187, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F2, using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- OOA 4-folding [i] based on linear OA(2209, 16432, F2, 29) (dual of [16432, 16223, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2209, 4108, F2, 4, 29) (dual of [(4108, 4), 16223, 30]-NRT-code), using
(209−29, 209, 179412)-Net in Base 2 — Upper bound on s
There is no (180, 209, 179413)-net in base 2, because
- 1 times m-reduction [i] would yield (180, 208, 179413)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 411 377951 180406 066184 835431 932056 283678 074877 754017 512397 285952 > 2208 [i]