Best Known (49, 49+20, s)-Nets in Base 7
(49, 49+20, 240)-Net over F7 — Constructive and digital
Digital (49, 69, 240)-net over F7, using
- net defined by OOA [i] based on linear OOA(769, 240, F7, 20, 20) (dual of [(240, 20), 4731, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(769, 2400, F7, 20) (dual of [2400, 2331, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(769, 2401, F7, 20) (dual of [2401, 2332, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(769, 2401, F7, 20) (dual of [2401, 2332, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(769, 2400, F7, 20) (dual of [2400, 2331, 21]-code), using
(49, 49+20, 1951)-Net over F7 — Digital
Digital (49, 69, 1951)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(769, 1951, F7, 20) (dual of [1951, 1882, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(769, 2401, F7, 20) (dual of [2401, 2332, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(769, 2401, F7, 20) (dual of [2401, 2332, 21]-code), using
(49, 49+20, 511677)-Net in Base 7 — Upper bound on s
There is no (49, 69, 511678)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 20500 726716 495190 476647 832818 513865 138458 588303 367959 435869 > 769 [i]