Best Known (111, 132, s)-Nets in Base 2
(111, 132, 820)-Net over F2 — Constructive and digital
Digital (111, 132, 820)-net over F2, using
- net defined by OOA [i] based on linear OOA(2132, 820, F2, 21, 21) (dual of [(820, 21), 17088, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2132, 8201, F2, 21) (dual of [8201, 8069, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2132, 8206, F2, 21) (dual of [8206, 8074, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2131, 8192, F2, 21) (dual of [8192, 8061, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2118, 8192, F2, 19) (dual of [8192, 8074, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2132, 8206, F2, 21) (dual of [8206, 8074, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2132, 8201, F2, 21) (dual of [8201, 8069, 22]-code), using
(111, 132, 1718)-Net over F2 — Digital
Digital (111, 132, 1718)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2132, 1718, F2, 4, 21) (dual of [(1718, 4), 6740, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2132, 2051, F2, 4, 21) (dual of [(2051, 4), 8072, 22]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2132, 8204, F2, 21) (dual of [8204, 8072, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2132, 8206, F2, 21) (dual of [8206, 8074, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2131, 8192, F2, 21) (dual of [8192, 8061, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2118, 8192, F2, 19) (dual of [8192, 8074, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2132, 8206, F2, 21) (dual of [8206, 8074, 22]-code), using
- OOA 4-folding [i] based on linear OA(2132, 8204, F2, 21) (dual of [8204, 8072, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(2132, 2051, F2, 4, 21) (dual of [(2051, 4), 8072, 22]-NRT-code), using
(111, 132, 39747)-Net in Base 2 — Upper bound on s
There is no (111, 132, 39748)-net in base 2, because
- 1 times m-reduction [i] would yield (111, 131, 39748)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2722 535364 661680 392323 513133 653073 108776 > 2131 [i]