Best Known (51, 68, s)-Nets in Base 7
(51, 68, 601)-Net over F7 — Constructive and digital
Digital (51, 68, 601)-net over F7, using
- net defined by OOA [i] based on linear OOA(768, 601, F7, 17, 17) (dual of [(601, 17), 10149, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(768, 4809, F7, 17) (dual of [4809, 4741, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(768, 4814, F7, 17) (dual of [4814, 4746, 18]-code), using
- trace code [i] based on linear OA(4934, 2407, F49, 17) (dual of [2407, 2373, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [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]
- linear OA(4929, 2402, F49, 15) (dual of [2402, 2373, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [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 C([0,8]) ⊂ C([0,7]) [i] based on
- trace code [i] based on linear OA(4934, 2407, F49, 17) (dual of [2407, 2373, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(768, 4814, F7, 17) (dual of [4814, 4746, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(768, 4809, F7, 17) (dual of [4809, 4741, 18]-code), using
(51, 68, 4879)-Net over F7 — Digital
Digital (51, 68, 4879)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(768, 4879, F7, 17) (dual of [4879, 4811, 18]-code), using
- 71 step Varšamov–Edel lengthening with (ri) = (1, 8 times 0, 1, 61 times 0) [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
- 71 step Varšamov–Edel lengthening with (ri) = (1, 8 times 0, 1, 61 times 0) [i] based on linear OA(766, 4806, F7, 17) (dual of [4806, 4740, 18]-code), using
(51, 68, 7502994)-Net in Base 7 — Upper bound on s
There is no (51, 68, 7502995)-net in base 7, because
- 1 times m-reduction [i] would yield (51, 67, 7502995)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 418 378129 617493 026384 883404 404687 671616 186425 191866 700625 > 767 [i]