Best Known (148−25, 148, s)-Nets in Base 2
(148−25, 148, 342)-Net over F2 — Constructive and digital
Digital (123, 148, 342)-net over F2, using
- 22 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
(148−25, 148, 1027)-Net over F2 — Digital
Digital (123, 148, 1027)-net over F2, using
- 22 times duplication [i] based on digital (121, 146, 1027)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2146, 1027, F2, 4, 25) (dual of [(1027, 4), 3962, 26]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2146, 4108, F2, 25) (dual of [4108, 3962, 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 4-folding [i] based on linear OA(2146, 4108, F2, 25) (dual of [4108, 3962, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2146, 1027, F2, 4, 25) (dual of [(1027, 4), 3962, 26]-NRT-code), using
(148−25, 148, 25744)-Net in Base 2 — Upper bound on s
There is no (123, 148, 25745)-net in base 2, because
- 1 times m-reduction [i] would yield (123, 147, 25745)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 178 450599 171125 784452 108298 369654 413305 302684 > 2147 [i]