Best Known (42, 56, s)-Nets in Base 7
(42, 56, 687)-Net over F7 — Constructive and digital
Digital (42, 56, 687)-net over F7, using
- net defined by OOA [i] based on linear OOA(756, 687, F7, 14, 14) (dual of [(687, 14), 9562, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(756, 4809, F7, 14) (dual of [4809, 4753, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(756, 4812, F7, 14) (dual of [4812, 4756, 15]-code), using
- trace code [i] based on linear OA(4928, 2406, F49, 14) (dual of [2406, 2378, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(4923, 2401, F49, 12) (dual of [2401, 2378, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(491, 5, F49, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(4928, 2406, F49, 14) (dual of [2406, 2378, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(756, 4812, F7, 14) (dual of [4812, 4756, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(756, 4809, F7, 14) (dual of [4809, 4753, 15]-code), using
(42, 56, 4855)-Net over F7 — Digital
Digital (42, 56, 4855)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(756, 4855, F7, 14) (dual of [4855, 4799, 15]-code), using
- 47 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0, 1, 40 times 0) [i] based on linear OA(754, 4806, F7, 14) (dual of [4806, 4752, 15]-code), using
- trace code [i] based on linear OA(4927, 2403, F49, 14) (dual of [2403, 2376, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(4927, 2401, F49, 14) (dual of [2401, 2374, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(4925, 2401, F49, 13) (dual of [2401, 2376, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(490, 2, F49, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(4927, 2403, F49, 14) (dual of [2403, 2376, 15]-code), using
- 47 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0, 1, 40 times 0) [i] based on linear OA(754, 4806, F7, 14) (dual of [4806, 4752, 15]-code), using
(42, 56, 3247514)-Net in Base 7 — Upper bound on s
There is no (42, 56, 3247515)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 211587 631327 564951 404935 688670 040273 851331 755639 > 756 [i]