Best Known (207, 207+25, s)-Nets in Base 2
(207, 207+25, 43692)-Net over F2 — Constructive and digital
Digital (207, 232, 43692)-net over F2, using
- 22 times duplication [i] based on digital (205, 230, 43692)-net over F2, using
- net defined by OOA [i] based on linear OOA(2230, 43692, F2, 25, 25) (dual of [(43692, 25), 1092070, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2230, 524305, F2, 25) (dual of [524305, 524075, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2230, 524308, F2, 25) (dual of [524308, 524078, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2229, 524288, F2, 25) (dual of [524288, 524059, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2210, 524288, F2, 23) (dual of [524288, 524078, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2230, 524308, F2, 25) (dual of [524308, 524078, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2230, 524305, F2, 25) (dual of [524305, 524075, 26]-code), using
- net defined by OOA [i] based on linear OOA(2230, 43692, F2, 25, 25) (dual of [(43692, 25), 1092070, 26]-NRT-code), using
(207, 207+25, 69185)-Net over F2 — Digital
Digital (207, 232, 69185)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2232, 69185, F2, 7, 25) (dual of [(69185, 7), 484063, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2232, 74901, F2, 7, 25) (dual of [(74901, 7), 524075, 26]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2230, 74901, F2, 7, 25) (dual of [(74901, 7), 524077, 26]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2230, 524307, F2, 25) (dual of [524307, 524077, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2230, 524308, F2, 25) (dual of [524308, 524078, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2229, 524288, F2, 25) (dual of [524288, 524059, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2210, 524288, F2, 23) (dual of [524288, 524078, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2230, 524308, F2, 25) (dual of [524308, 524078, 26]-code), using
- OOA 7-folding [i] based on linear OA(2230, 524307, F2, 25) (dual of [524307, 524077, 26]-code), using
- 22 times duplication [i] based on linear OOA(2230, 74901, F2, 7, 25) (dual of [(74901, 7), 524077, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2232, 74901, F2, 7, 25) (dual of [(74901, 7), 524075, 26]-NRT-code), using
(207, 207+25, 3297513)-Net in Base 2 — Upper bound on s
There is no (207, 232, 3297514)-net in base 2, because
- 1 times m-reduction [i] would yield (207, 231, 3297514)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3450 884870 051221 939877 019818 441015 295530 462324 571615 422430 473896 963272 > 2231 [i]