Best Known (81−13, 81, s)-Nets in Base 2
(81−13, 81, 1367)-Net over F2 — Constructive and digital
Digital (68, 81, 1367)-net over F2, using
- 21 times duplication [i] based on digital (67, 80, 1367)-net over F2, using
- net defined by OOA [i] based on linear OOA(280, 1367, F2, 13, 13) (dual of [(1367, 13), 17691, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(280, 8203, F2, 13) (dual of [8203, 8123, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(280, 8206, F2, 13) (dual of [8206, 8126, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(279, 8192, F2, 13) (dual of [8192, 8113, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(266, 8192, F2, 11) (dual of [8192, 8126, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(280, 8206, F2, 13) (dual of [8206, 8126, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(280, 8203, F2, 13) (dual of [8203, 8123, 14]-code), using
- net defined by OOA [i] based on linear OOA(280, 1367, F2, 13, 13) (dual of [(1367, 13), 17691, 14]-NRT-code), using
(81−13, 81, 2052)-Net over F2 — Digital
Digital (68, 81, 2052)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(281, 2052, F2, 4, 13) (dual of [(2052, 4), 8127, 14]-NRT-code), using
- OOA 4-folding [i] based on linear OA(281, 8208, F2, 13) (dual of [8208, 8127, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(280, 8207, F2, 13) (dual of [8207, 8127, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(279, 8192, F2, 13) (dual of [8192, 8113, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(266, 8192, F2, 11) (dual of [8192, 8126, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(214, 15, F2, 13) (dual of [15, 1, 14]-code), using
- strength reduction [i] based on linear OA(214, 15, F2, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
- dual of repetition code with length 15 [i]
- strength reduction [i] based on linear OA(214, 15, F2, 14) (dual of [15, 1, 15]-code or 15-arc in PG(13,2)), using
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(280, 8207, F2, 13) (dual of [8207, 8127, 14]-code), using
- OOA 4-folding [i] based on linear OA(281, 8208, F2, 13) (dual of [8208, 8127, 14]-code), using
(81−13, 81, 30891)-Net in Base 2 — Upper bound on s
There is no (68, 81, 30892)-net in base 2, because
- 1 times m-reduction [i] would yield (68, 80, 30892)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 209095 092503 258853 684310 > 280 [i]