Best Known (103, 103+20, s)-Nets in Base 2
(103, 103+20, 411)-Net over F2 — Constructive and digital
Digital (103, 123, 411)-net over F2, using
- net defined by OOA [i] based on linear OOA(2123, 411, F2, 20, 20) (dual of [(411, 20), 8097, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(2123, 4110, F2, 20) (dual of [4110, 3987, 21]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2121, 4108, F2, 20) (dual of [4108, 3987, 21]-code), using
- 1 times truncation [i] based on linear OA(2122, 4109, F2, 21) (dual of [4109, 3987, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2121, 4096, F2, 21) (dual of [4096, 3975, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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
- 1 times truncation [i] based on linear OA(2122, 4109, F2, 21) (dual of [4109, 3987, 22]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2121, 4108, F2, 20) (dual of [4108, 3987, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(2123, 4110, F2, 20) (dual of [4110, 3987, 21]-code), using
(103, 103+20, 1208)-Net over F2 — Digital
Digital (103, 123, 1208)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2123, 1208, F2, 3, 20) (dual of [(1208, 3), 3501, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2123, 1370, F2, 3, 20) (dual of [(1370, 3), 3987, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2123, 4110, F2, 20) (dual of [4110, 3987, 21]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2121, 4108, F2, 20) (dual of [4108, 3987, 21]-code), using
- 1 times truncation [i] based on linear OA(2122, 4109, F2, 21) (dual of [4109, 3987, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2121, 4096, F2, 21) (dual of [4096, 3975, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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
- 1 times truncation [i] based on linear OA(2122, 4109, F2, 21) (dual of [4109, 3987, 22]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2121, 4108, F2, 20) (dual of [4108, 3987, 21]-code), using
- OOA 3-folding [i] based on linear OA(2123, 4110, F2, 20) (dual of [4110, 3987, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(2123, 1370, F2, 3, 20) (dual of [(1370, 3), 3987, 21]-NRT-code), using
(103, 103+20, 22822)-Net in Base 2 — Upper bound on s
There is no (103, 123, 22823)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 10 634615 373474 369859 488419 723990 324076 > 2123 [i]