Best Known (84−19, 84, s)-Nets in Base 2
(84−19, 84, 135)-Net over F2 — Constructive and digital
Digital (65, 84, 135)-net over F2, using
- trace code for nets [i] based on digital (9, 28, 45)-net over F8, using
- net from sequence [i] based on digital (9, 44)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 9 and N(F) ≥ 45, using
- net from sequence [i] based on digital (9, 44)-sequence over F8, using
(84−19, 84, 216)-Net over F2 — Digital
Digital (65, 84, 216)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(284, 216, F2, 2, 19) (dual of [(216, 2), 348, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(284, 265, F2, 2, 19) (dual of [(265, 2), 446, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(284, 530, F2, 19) (dual of [530, 446, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(284, 531, F2, 19) (dual of [531, 447, 20]-code), using
- construction XX applied to C1 = C([509,14]), C2 = C([0,16]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([509,16]) [i] based on
- linear OA(273, 511, F2, 17) (dual of [511, 438, 18]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,14}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(273, 511, F2, 17) (dual of [511, 438, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(282, 511, F2, 19) (dual of [511, 429, 20]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,16}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(264, 511, F2, 15) (dual of [511, 447, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 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, using
- linear OA(21, 10, F2, 1) (dual of [10, 9, 2]-code) (see above)
- construction XX applied to C1 = C([509,14]), C2 = C([0,16]), C3 = C1 + C2 = C([0,14]), and C∩ = C1 ∩ C2 = C([509,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(284, 531, F2, 19) (dual of [531, 447, 20]-code), using
- OOA 2-folding [i] based on linear OA(284, 530, F2, 19) (dual of [530, 446, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(284, 265, F2, 2, 19) (dual of [(265, 2), 446, 20]-NRT-code), using
(84−19, 84, 2463)-Net in Base 2 — Upper bound on s
There is no (65, 84, 2464)-net in base 2, because
- 1 times m-reduction [i] would yield (65, 83, 2464)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 9 672153 122456 589961 826213 > 283 [i]