Best Known (67, 67+18, s)-Nets in Base 2
(67, 67+18, 152)-Net over F2 — Constructive and digital
Digital (67, 85, 152)-net over F2, using
- 21 times duplication [i] based on digital (66, 84, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 21, 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, 21, 38)-net over F16, using
(67, 67+18, 266)-Net over F2 — Digital
Digital (67, 85, 266)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(285, 266, F2, 2, 18) (dual of [(266, 2), 447, 19]-NRT-code), using
- strength reduction [i] based on linear OOA(285, 266, F2, 2, 19) (dual of [(266, 2), 447, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(285, 532, F2, 19) (dual of [532, 447, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(285, 533, F2, 19) (dual of [533, 448, 20]-code), using
- adding a parity check bit [i] based on linear OA(284, 532, F2, 18) (dual of [532, 448, 19]-code), using
- construction XX applied to C1 = C([509,14]), C2 = C([1,16]), C3 = C1 + C2 = C([1,14]), 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(272, 511, F2, 16) (dual of [511, 439, 17]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(263, 511, F2, 14) (dual of [511, 448, 15]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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([509,14]), C2 = C([1,16]), C3 = C1 + C2 = C([1,14]), and C∩ = C1 ∩ C2 = C([509,16]) [i] based on
- adding a parity check bit [i] based on linear OA(284, 532, F2, 18) (dual of [532, 448, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(285, 533, F2, 19) (dual of [533, 448, 20]-code), using
- OOA 2-folding [i] based on linear OA(285, 532, F2, 19) (dual of [532, 447, 20]-code), using
- strength reduction [i] based on linear OOA(285, 266, F2, 2, 19) (dual of [(266, 2), 447, 20]-NRT-code), using
(67, 67+18, 2876)-Net in Base 2 — Upper bound on s
There is no (67, 85, 2877)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 38 750952 144477 694205 237166 > 285 [i]