Best Known (115−25, 115, s)-Nets in Base 2
(115−25, 115, 152)-Net over F2 — Constructive and digital
Digital (90, 115, 152)-net over F2, using
- 1 times m-reduction [i] based on digital (90, 116, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 29, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- trace code for nets [i] based on digital (3, 29, 38)-net over F16, using
(115−25, 115, 268)-Net over F2 — Digital
Digital (90, 115, 268)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2115, 268, F2, 2, 25) (dual of [(268, 2), 421, 26]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2113, 267, F2, 2, 25) (dual of [(267, 2), 421, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2113, 534, F2, 25) (dual of [534, 421, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2112, 533, F2, 25) (dual of [533, 421, 26]-code), using
- adding a parity check bit [i] based on linear OA(2111, 532, F2, 24) (dual of [532, 421, 25]-code), using
- construction XX applied to C1 = C([509,20]), C2 = C([1,22]), C3 = C1 + C2 = C([1,20]), and C∩ = C1 ∩ C2 = C([509,22]) [i] based on
- linear OA(2100, 511, F2, 23) (dual of [511, 411, 24]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,20}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(299, 511, F2, 22) (dual of [511, 412, 23]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2109, 511, F2, 25) (dual of [511, 402, 26]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,22}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(290, 511, F2, 20) (dual of [511, 421, 21]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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 (see above)
- construction XX applied to C1 = C([509,20]), C2 = C([1,22]), C3 = C1 + C2 = C([1,20]), and C∩ = C1 ∩ C2 = C([509,22]) [i] based on
- adding a parity check bit [i] based on linear OA(2111, 532, F2, 24) (dual of [532, 421, 25]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2112, 533, F2, 25) (dual of [533, 421, 26]-code), using
- OOA 2-folding [i] based on linear OA(2113, 534, F2, 25) (dual of [534, 421, 26]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2113, 267, F2, 2, 25) (dual of [(267, 2), 421, 26]-NRT-code), using
(115−25, 115, 3812)-Net in Base 2 — Upper bound on s
There is no (90, 115, 3813)-net in base 2, because
- 1 times m-reduction [i] would yield (90, 114, 3813)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 20829 543713 330682 763565 330357 464284 > 2114 [i]