Best Known (61, 61+20, s)-Nets in Base 7
(61, 61+20, 481)-Net over F7 — Constructive and digital
Digital (61, 81, 481)-net over F7, using
- 71 times duplication [i] based on digital (60, 80, 481)-net over F7, using
- net defined by OOA [i] based on linear OOA(780, 481, F7, 20, 20) (dual of [(481, 20), 9540, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(780, 4810, F7, 20) (dual of [4810, 4730, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(780, 4812, F7, 20) (dual of [4812, 4732, 21]-code), using
- trace code [i] based on linear OA(4940, 2406, F49, 20) (dual of [2406, 2366, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- 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(4935, 2401, F49, 18) (dual of [2401, 2366, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(19) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(4940, 2406, F49, 20) (dual of [2406, 2366, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(780, 4812, F7, 20) (dual of [4812, 4732, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(780, 4810, F7, 20) (dual of [4810, 4730, 21]-code), using
- net defined by OOA [i] based on linear OOA(780, 481, F7, 20, 20) (dual of [(481, 20), 9540, 21]-NRT-code), using
(61, 61+20, 5343)-Net over F7 — Digital
Digital (61, 81, 5343)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(781, 5343, F7, 20) (dual of [5343, 5262, 21]-code), using
- 534 step Varšamov–Edel lengthening with (ri) = (1, 26 times 0, 1, 137 times 0, 1, 368 times 0) [i] based on linear OA(778, 4806, F7, 20) (dual of [4806, 4728, 21]-code), using
- trace code [i] based on linear OA(4939, 2403, F49, 20) (dual of [2403, 2364, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- 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(4937, 2401, F49, 19) (dual of [2401, 2364, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- trace code [i] based on linear OA(4939, 2403, F49, 20) (dual of [2403, 2364, 21]-code), using
- 534 step Varšamov–Edel lengthening with (ri) = (1, 26 times 0, 1, 137 times 0, 1, 368 times 0) [i] based on linear OA(778, 4806, F7, 20) (dual of [4806, 4728, 21]-code), using
(61, 61+20, 5285896)-Net in Base 7 — Upper bound on s
There is no (61, 81, 5285897)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 283 753678 640414 630879 740930 448751 245517 578465 540149 116909 984286 076161 > 781 [i]