Best Known (82−21, 82, s)-Nets in Base 7
(82−21, 82, 480)-Net over F7 — Constructive and digital
Digital (61, 82, 480)-net over F7, using
- net defined by OOA [i] based on linear OOA(782, 480, F7, 21, 21) (dual of [(480, 21), 9998, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(782, 4801, F7, 21) (dual of [4801, 4719, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(782, 4804, F7, 21) (dual of [4804, 4722, 22]-code), using
- trace code [i] based on linear OA(4941, 2402, F49, 21) (dual of [2402, 2361, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- trace code [i] based on linear OA(4941, 2402, F49, 21) (dual of [2402, 2361, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(782, 4804, F7, 21) (dual of [4804, 4722, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(782, 4801, F7, 21) (dual of [4801, 4719, 22]-code), using
(82−21, 82, 4806)-Net over F7 — Digital
Digital (61, 82, 4806)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(782, 4806, F7, 21) (dual of [4806, 4724, 22]-code), using
- trace code [i] based on linear OA(4941, 2403, F49, 21) (dual of [2403, 2362, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(4941, 2401, F49, 21) (dual of [2401, 2360, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4939, 2401, F49, 20) (dual of [2401, 2362, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(20) ⊂ Ce(19) [i] based on
- trace code [i] based on linear OA(4941, 2403, F49, 21) (dual of [2403, 2362, 22]-code), using
(82−21, 82, 5285896)-Net in Base 7 — Upper bound on s
There is no (61, 82, 5285897)-net in base 7, because
- 1 times m-reduction [i] would yield (61, 81, 5285897)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 283 753678 640414 630879 740930 448751 245517 578465 540149 116909 984286 076161 > 781 [i]