Best Known (49, 49+17, s)-Nets in Base 7
(49, 49+17, 600)-Net over F7 — Constructive and digital
Digital (49, 66, 600)-net over F7, using
- net defined by OOA [i] based on linear OOA(766, 600, F7, 17, 17) (dual of [(600, 17), 10134, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(766, 4801, F7, 17) (dual of [4801, 4735, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(766, 4804, F7, 17) (dual of [4804, 4738, 18]-code), using
- trace code [i] based on linear OA(4933, 2402, F49, 17) (dual of [2402, 2369, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- trace code [i] based on linear OA(4933, 2402, F49, 17) (dual of [2402, 2369, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(766, 4804, F7, 17) (dual of [4804, 4738, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(766, 4801, F7, 17) (dual of [4801, 4735, 18]-code), using
(49, 49+17, 4806)-Net over F7 — Digital
Digital (49, 66, 4806)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(766, 4806, F7, 17) (dual of [4806, 4740, 18]-code), using
- trace code [i] based on linear OA(4933, 2403, F49, 17) (dual of [2403, 2370, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(4933, 2401, F49, 17) (dual of [2401, 2368, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(4931, 2401, F49, 16) (dual of [2401, 2370, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(16) ⊂ Ce(15) [i] based on
- trace code [i] based on linear OA(4933, 2403, F49, 17) (dual of [2403, 2370, 18]-code), using
(49, 49+17, 4612750)-Net in Base 7 — Upper bound on s
There is no (49, 66, 4612751)-net in base 7, because
- 1 times m-reduction [i] would yield (49, 65, 4612751)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 8 538335 457563 632308 382226 127437 257511 657175 202626 709201 > 765 [i]