Best Known (18, 25, s)-Nets in Base 7
(18, 25, 801)-Net over F7 — Constructive and digital
Digital (18, 25, 801)-net over F7, using
- net defined by OOA [i] based on linear OOA(725, 801, F7, 7, 7) (dual of [(801, 7), 5582, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(725, 2404, F7, 7) (dual of [2404, 2379, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(725, 2405, F7, 7) (dual of [2405, 2380, 8]-code), using
- 1 times truncation [i] based on linear OA(726, 2406, F7, 8) (dual of [2406, 2380, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(721, 2401, F7, 6) (dual of [2401, 2380, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(71, 5, F7, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- 1 times truncation [i] based on linear OA(726, 2406, F7, 8) (dual of [2406, 2380, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(725, 2405, F7, 7) (dual of [2405, 2380, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(725, 2404, F7, 7) (dual of [2404, 2379, 8]-code), using
(18, 25, 2436)-Net over F7 — Digital
Digital (18, 25, 2436)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(725, 2436, F7, 7) (dual of [2436, 2411, 8]-code), using
- 35 step Varšamov–Edel lengthening with (ri) = (1, 34 times 0) [i] based on linear OA(724, 2400, F7, 7) (dual of [2400, 2376, 8]-code), using
- 1 times truncation [i] based on linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 1 times truncation [i] based on linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using
- 35 step Varšamov–Edel lengthening with (ri) = (1, 34 times 0) [i] based on linear OA(724, 2400, F7, 7) (dual of [2400, 2376, 8]-code), using
(18, 25, 1745888)-Net in Base 7 — Upper bound on s
There is no (18, 25, 1745889)-net in base 7, because
- 1 times m-reduction [i] would yield (18, 24, 1745889)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 191 581472 206541 353303 > 724 [i]