Best Known (68, 87, s)-Nets in Base 2
(68, 87, 138)-Net over F2 — Constructive and digital
Digital (68, 87, 138)-net over F2, using
- trace code for nets [i] based on digital (10, 29, 46)-net over F8, using
- net from sequence [i] based on digital (10, 45)-sequence over F8, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F8 with g(F) = 9, N(F) = 45, and 1 place with degree 2 [i] based on function field F/F8 with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (10, 45)-sequence over F8, using
(68, 87, 248)-Net over F2 — Digital
Digital (68, 87, 248)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(287, 248, F2, 2, 19) (dual of [(248, 2), 409, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(287, 267, F2, 2, 19) (dual of [(267, 2), 447, 20]-NRT-code), using
- 21 times duplication [i] based on linear OOA(286, 267, F2, 2, 19) (dual of [(267, 2), 448, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(286, 534, F2, 19) (dual of [534, 448, 20]-code), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on linear OA(285, 533, F2, 19) (dual of [533, 448, 20]-code), using
- OOA 2-folding [i] based on linear OA(286, 534, F2, 19) (dual of [534, 448, 20]-code), using
- 21 times duplication [i] based on linear OOA(286, 267, F2, 2, 19) (dual of [(267, 2), 448, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(287, 267, F2, 2, 19) (dual of [(267, 2), 447, 20]-NRT-code), using
(68, 87, 3107)-Net in Base 2 — Upper bound on s
There is no (68, 87, 3108)-net in base 2, because
- 1 times m-reduction [i] would yield (68, 86, 3108)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 77 419563 814467 314091 704252 > 286 [i]