Best Known (241−23, 241, s)-Nets in Base 2
(241−23, 241, 190654)-Net over F2 — Constructive and digital
Digital (218, 241, 190654)-net over F2, using
- 22 times duplication [i] based on digital (216, 239, 190654)-net over F2, using
- net defined by OOA [i] based on linear OOA(2239, 190654, F2, 23, 23) (dual of [(190654, 23), 4384803, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2239, 2097195, F2, 23) (dual of [2097195, 2096956, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2239, 2097201, F2, 23) (dual of [2097201, 2096962, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2232, 2097152, F2, 23) (dual of [2097152, 2096920, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(27, 49, F2, 3) (dual of [49, 42, 4]-code or 49-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 Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2239, 2097201, F2, 23) (dual of [2097201, 2096962, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2239, 2097195, F2, 23) (dual of [2097195, 2096956, 24]-code), using
- net defined by OOA [i] based on linear OOA(2239, 190654, F2, 23, 23) (dual of [(190654, 23), 4384803, 24]-NRT-code), using
(241−23, 241, 299600)-Net over F2 — Digital
Digital (218, 241, 299600)-net over F2, using
- 21 times duplication [i] based on digital (217, 240, 299600)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2240, 299600, F2, 7, 23) (dual of [(299600, 7), 2096960, 24]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2239, 299600, F2, 7, 23) (dual of [(299600, 7), 2096961, 24]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2239, 2097200, F2, 23) (dual of [2097200, 2096961, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2239, 2097201, F2, 23) (dual of [2097201, 2096962, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2232, 2097152, F2, 23) (dual of [2097152, 2096920, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(27, 49, F2, 3) (dual of [49, 42, 4]-code or 49-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 Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2239, 2097201, F2, 23) (dual of [2097201, 2096962, 24]-code), using
- OOA 7-folding [i] based on linear OA(2239, 2097200, F2, 23) (dual of [2097200, 2096961, 24]-code), using
- 21 times duplication [i] based on linear OOA(2239, 299600, F2, 7, 23) (dual of [(299600, 7), 2096961, 24]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2240, 299600, F2, 7, 23) (dual of [(299600, 7), 2096960, 24]-NRT-code), using
(241−23, 241, large)-Net in Base 2 — Upper bound on s
There is no (218, 241, large)-net in base 2, because
- 21 times m-reduction [i] would yield (218, 220, large)-net in base 2, but