Best Known (27−7, 27, s)-Nets in Base 7
(27−7, 27, 1602)-Net over F7 — Constructive and digital
Digital (20, 27, 1602)-net over F7, using
- net defined by OOA [i] based on linear OOA(727, 1602, F7, 7, 7) (dual of [(1602, 7), 11187, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(727, 4807, F7, 7) (dual of [4807, 4780, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(726, 4806, F7, 7) (dual of [4806, 4780, 8]-code), using
- trace code [i] based on linear OA(4913, 2403, F49, 7) (dual of [2403, 2390, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(4913, 2401, F49, 7) (dual of [2401, 2388, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(4911, 2401, F49, 6) (dual of [2401, 2390, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(4913, 2403, F49, 7) (dual of [2403, 2390, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(726, 4806, F7, 7) (dual of [4806, 4780, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(727, 4807, F7, 7) (dual of [4807, 4780, 8]-code), using
(27−7, 27, 4862)-Net over F7 — Digital
Digital (20, 27, 4862)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(727, 4862, F7, 7) (dual of [4862, 4835, 8]-code), using
- 55 step Varšamov–Edel lengthening with (ri) = (1, 54 times 0) [i] based on linear OA(726, 4806, F7, 7) (dual of [4806, 4780, 8]-code), using
- trace code [i] based on linear OA(4913, 2403, F49, 7) (dual of [2403, 2390, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(4913, 2401, F49, 7) (dual of [2401, 2388, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(4911, 2401, F49, 6) (dual of [2401, 2390, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(4913, 2403, F49, 7) (dual of [2403, 2390, 8]-code), using
- 55 step Varšamov–Edel lengthening with (ri) = (1, 54 times 0) [i] based on linear OA(726, 4806, F7, 7) (dual of [4806, 4780, 8]-code), using
(27−7, 27, 6388742)-Net in Base 7 — Upper bound on s
There is no (20, 27, 6388743)-net in base 7, because
- 1 times m-reduction [i] would yield (20, 26, 6388743)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 9387 480783 983248 929583 > 726 [i]