Best Known (43, 57, s)-Nets in Base 7
(43, 57, 687)-Net over F7 — Constructive and digital
Digital (43, 57, 687)-net over F7, using
- 71 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(756, 687, F7, 14, 14) (dual of [(687, 14), 9562, 15]-NRT-code), using
(43, 57, 5066)-Net over F7 — Digital
Digital (43, 57, 5066)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(757, 5066, F7, 14) (dual of [5066, 5009, 15]-code), using
- 257 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0, 1, 40 times 0, 1, 209 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
- 257 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0, 1, 40 times 0, 1, 209 times 0) [i] based on linear OA(754, 4806, F7, 14) (dual of [4806, 4752, 15]-code), using
(43, 57, 4288244)-Net in Base 7 — Upper bound on s
There is no (43, 57, 4288245)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1 481113 360266 196038 307038 742119 489617 585422 629655 > 757 [i]