Best Known (79−12, 79, s)-Nets in Base 2
(79−12, 79, 1367)-Net over F2 — Constructive and digital
Digital (67, 79, 1367)-net over F2, using
- net defined by OOA [i] based on linear OOA(279, 1367, F2, 12, 12) (dual of [(1367, 12), 16325, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(279, 8202, F2, 12) (dual of [8202, 8123, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(279, 8206, F2, 12) (dual of [8206, 8127, 13]-code), using
- 1 times truncation [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 truncation [i] based on linear OA(280, 8207, F2, 13) (dual of [8207, 8127, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(279, 8206, F2, 12) (dual of [8206, 8127, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(279, 8202, F2, 12) (dual of [8202, 8123, 13]-code), using
(79−12, 79, 2715)-Net over F2 — Digital
Digital (67, 79, 2715)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(279, 2715, F2, 3, 12) (dual of [(2715, 3), 8066, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(279, 2735, F2, 3, 12) (dual of [(2735, 3), 8126, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(279, 8205, F2, 12) (dual of [8205, 8126, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(279, 8206, F2, 12) (dual of [8206, 8127, 13]-code), using
- 1 times truncation [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 truncation [i] based on linear OA(280, 8207, F2, 13) (dual of [8207, 8127, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(279, 8206, F2, 12) (dual of [8206, 8127, 13]-code), using
- OOA 3-folding [i] based on linear OA(279, 8205, F2, 12) (dual of [8205, 8126, 13]-code), using
- discarding factors / shortening the dual code based on linear OOA(279, 2735, F2, 3, 12) (dual of [(2735, 3), 8126, 13]-NRT-code), using
(79−12, 79, 27520)-Net in Base 2 — Upper bound on s
There is no (67, 79, 27521)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 604584 957468 107433 910912 > 279 [i]