Best Known (18, 24, s)-Nets in Base 7
(18, 24, 1604)-Net over F7 — Constructive and digital
Digital (18, 24, 1604)-net over F7, using
- net defined by OOA [i] based on linear OOA(724, 1604, F7, 6, 6) (dual of [(1604, 6), 9600, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(724, 4812, F7, 6) (dual of [4812, 4788, 7]-code), using
- trace code [i] based on linear OA(4912, 2406, F49, 6) (dual of [2406, 2394, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- 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(497, 2401, F49, 4) (dual of [2401, 2394, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(5) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(4912, 2406, F49, 6) (dual of [2406, 2394, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(724, 4812, F7, 6) (dual of [4812, 4788, 7]-code), using
(18, 24, 5592)-Net over F7 — Digital
Digital (18, 24, 5592)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(724, 5592, F7, 6) (dual of [5592, 5568, 7]-code), using
- 784 step Varšamov–Edel lengthening with (ri) = (1, 127 times 0, 1, 655 times 0) [i] based on linear OA(722, 4806, F7, 6) (dual of [4806, 4784, 7]-code), using
- trace code [i] based on linear OA(4911, 2403, F49, 6) (dual of [2403, 2392, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- 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(499, 2401, F49, 5) (dual of [2401, 2392, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- trace code [i] based on linear OA(4911, 2403, F49, 6) (dual of [2403, 2392, 7]-code), using
- 784 step Varšamov–Edel lengthening with (ri) = (1, 127 times 0, 1, 655 times 0) [i] based on linear OA(722, 4806, F7, 6) (dual of [4806, 4784, 7]-code), using
(18, 24, 1745888)-Net in Base 7 — Upper bound on s
There is no (18, 24, 1745889)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 191 581472 206541 353303 > 724 [i]