Best Known (84, 104, s)-Nets in Base 2
(84, 104, 260)-Net over F2 — Constructive and digital
Digital (84, 104, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 26, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
(84, 104, 436)-Net over F2 — Digital
Digital (84, 104, 436)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2104, 436, F2, 2, 20) (dual of [(436, 2), 768, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2104, 523, F2, 2, 20) (dual of [(523, 2), 942, 21]-NRT-code), using
- strength reduction [i] based on linear OOA(2104, 523, F2, 2, 21) (dual of [(523, 2), 942, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2104, 1046, F2, 21) (dual of [1046, 942, 22]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2104, 1047, F2, 21) (dual of [1047, 943, 22]-code), using
- OOA 2-folding [i] based on linear OA(2104, 1046, F2, 21) (dual of [1046, 942, 22]-code), using
- strength reduction [i] based on linear OOA(2104, 523, F2, 2, 21) (dual of [(523, 2), 942, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2104, 523, F2, 2, 20) (dual of [(523, 2), 942, 21]-NRT-code), using
(84, 104, 6104)-Net in Base 2 — Upper bound on s
There is no (84, 104, 6105)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 20 294831 788253 890323 654172 972724 > 2104 [i]