Best Known (124, 124+25, s)-Nets in Base 2
(124, 124+25, 342)-Net over F2 — Constructive and digital
Digital (124, 149, 342)-net over F2, using
- 23 times duplication [i] based on digital (121, 146, 342)-net over F2, using
- net defined by OOA [i] based on linear OOA(2146, 342, F2, 25, 25) (dual of [(342, 25), 8404, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2146, 4105, F2, 25) (dual of [4105, 3959, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2146, 4109, F2, 25) (dual of [4109, 3963, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2145, 4096, F2, 25) (dual of [4096, 3951, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2133, 4096, F2, 23) (dual of [4096, 3963, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2146, 4109, F2, 25) (dual of [4109, 3963, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2146, 4105, F2, 25) (dual of [4105, 3959, 26]-code), using
- net defined by OOA [i] based on linear OOA(2146, 342, F2, 25, 25) (dual of [(342, 25), 8404, 26]-NRT-code), using
(124, 124+25, 1045)-Net over F2 — Digital
Digital (124, 149, 1045)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2149, 1045, F2, 3, 25) (dual of [(1045, 3), 2986, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2149, 1370, F2, 3, 25) (dual of [(1370, 3), 3961, 26]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2147, 1370, F2, 3, 25) (dual of [(1370, 3), 3963, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2147, 4110, F2, 25) (dual of [4110, 3963, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2146, 4109, F2, 25) (dual of [4109, 3963, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2145, 4096, F2, 25) (dual of [4096, 3951, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2133, 4096, F2, 23) (dual of [4096, 3963, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2146, 4109, F2, 25) (dual of [4109, 3963, 26]-code), using
- OOA 3-folding [i] based on linear OA(2147, 4110, F2, 25) (dual of [4110, 3963, 26]-code), using
- 22 times duplication [i] based on linear OOA(2147, 1370, F2, 3, 25) (dual of [(1370, 3), 3963, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2149, 1370, F2, 3, 25) (dual of [(1370, 3), 3961, 26]-NRT-code), using
(124, 124+25, 27276)-Net in Base 2 — Upper bound on s
There is no (124, 149, 27277)-net in base 2, because
- 1 times m-reduction [i] would yield (124, 148, 27277)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 356 914167 351260 613502 909685 109859 017454 153937 > 2148 [i]