Best Known (216−21, 216, s)-Nets in Base 2
(216−21, 216, 209717)-Net over F2 — Constructive and digital
Digital (195, 216, 209717)-net over F2, using
- 24 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(2212, 209717, F2, 21, 21) (dual of [(209717, 21), 4403845, 22]-NRT-code), using
(216−21, 216, 299597)-Net over F2 — Digital
Digital (195, 216, 299597)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2216, 299597, F2, 7, 21) (dual of [(299597, 7), 2096963, 22]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2216, 2097179, F2, 21) (dual of [2097179, 2096963, 22]-code), using
- 4 times code embedding in larger space [i] based on linear OA(2212, 2097175, F2, 21) (dual of [2097175, 2096963, 22]-code), using
- construction X4 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(222, 23, F2, 21) (dual of [23, 1, 22]-code), using
- strength reduction [i] based on linear OA(222, 23, F2, 22) (dual of [23, 1, 23]-code or 23-arc in PG(21,2)), using
- dual of repetition code with length 23 [i]
- strength reduction [i] based on linear OA(222, 23, F2, 22) (dual of [23, 1, 23]-code or 23-arc in PG(21,2)), using
- linear OA(21, 23, F2, 1) (dual of [23, 22, 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 X4 applied to Ce(20) ⊂ Ce(18) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(2212, 2097175, F2, 21) (dual of [2097175, 2096963, 22]-code), using
- OOA 7-folding [i] based on linear OA(2216, 2097179, F2, 21) (dual of [2097179, 2096963, 22]-code), using
(216−21, 216, large)-Net in Base 2 — Upper bound on s
There is no (195, 216, large)-net in base 2, because
- 19 times m-reduction [i] would yield (195, 197, large)-net in base 2, but