Best Known (85, 85+21, s)-Nets in Base 2
(85, 85+21, 196)-Net over F2 — Constructive and digital
Digital (85, 106, 196)-net over F2, using
- 22 times duplication [i] based on digital (83, 104, 196)-net over F2, using
- trace code for nets [i] based on digital (5, 26, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- trace code for nets [i] based on digital (5, 26, 49)-net over F16, using
(85, 85+21, 389)-Net over F2 — Digital
Digital (85, 106, 389)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2106, 389, F2, 2, 21) (dual of [(389, 2), 672, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2106, 524, F2, 2, 21) (dual of [(524, 2), 942, 22]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2105, 524, F2, 2, 21) (dual of [(524, 2), 943, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2105, 1048, F2, 21) (dual of [1048, 943, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2104, 1047, F2, 21) (dual of [1047, 943, 22]-code), using
- adding a parity check bit [i] based on linear OA(2103, 1046, F2, 20) (dual of [1046, 943, 21]-code), using
- construction XX applied to C1 = C([1021,16]), C2 = C([1,18]), C3 = C1 + C2 = C([1,16]), and C∩ = C1 ∩ C2 = C([1021,18]) [i] based on
- linear OA(291, 1023, F2, 19) (dual of [1023, 932, 20]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,16}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(290, 1023, F2, 18) (dual of [1023, 933, 19]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2101, 1023, F2, 21) (dual of [1023, 922, 22]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,18}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(280, 1023, F2, 16) (dual of [1023, 943, 17]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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, 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 (see above)
- construction XX applied to C1 = C([1021,16]), C2 = C([1,18]), C3 = C1 + C2 = C([1,16]), and C∩ = C1 ∩ C2 = C([1021,18]) [i] based on
- adding a parity check bit [i] based on linear OA(2103, 1046, F2, 20) (dual of [1046, 943, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2104, 1047, F2, 21) (dual of [1047, 943, 22]-code), using
- OOA 2-folding [i] based on linear OA(2105, 1048, F2, 21) (dual of [1048, 943, 22]-code), using
- 21 times duplication [i] based on linear OOA(2105, 524, F2, 2, 21) (dual of [(524, 2), 943, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2106, 524, F2, 2, 21) (dual of [(524, 2), 942, 22]-NRT-code), using
(85, 85+21, 6543)-Net in Base 2 — Upper bound on s
There is no (85, 106, 6544)-net in base 2, because
- 1 times m-reduction [i] would yield (85, 105, 6544)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 40 576324 357443 369879 960693 388779 > 2105 [i]