Best Known (212−21, 212, s)-Nets in Base 2
(212−21, 212, 209717)-Net over F2 — Constructive and digital
Digital (191, 212, 209717)-net over F2, using
- net defined by OOA [i] based on linear OOA(2212, 209717, F2, 21, 21) (dual of [(209717, 21), 4403845, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2212, 2097171, F2, 21) (dual of [2097171, 2096959, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2212, 2097174, F2, 21) (dual of [2097174, 2096962, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2211, 2097152, F2, 21) (dual of [2097152, 2096941, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2190, 2097152, F2, 19) (dual of [2097152, 2096962, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 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(2212, 2097174, F2, 21) (dual of [2097174, 2096962, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2212, 2097171, F2, 21) (dual of [2097171, 2096959, 22]-code), using
(212−21, 212, 299596)-Net over F2 — Digital
Digital (191, 212, 299596)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2212, 299596, F2, 7, 21) (dual of [(299596, 7), 2096960, 22]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2212, 2097172, F2, 21) (dual of [2097172, 2096960, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2212, 2097174, F2, 21) (dual of [2097174, 2096962, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2211, 2097152, F2, 21) (dual of [2097152, 2096941, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2190, 2097152, F2, 19) (dual of [2097152, 2096962, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 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(2212, 2097174, F2, 21) (dual of [2097174, 2096962, 22]-code), using
- OOA 7-folding [i] based on linear OA(2212, 2097172, F2, 21) (dual of [2097172, 2096960, 22]-code), using
(212−21, 212, large)-Net in Base 2 — Upper bound on s
There is no (191, 212, large)-net in base 2, because
- 19 times m-reduction [i] would yield (191, 193, large)-net in base 2, but