Best Known (39, 39+13, s)-Nets in Base 7
(39, 39+13, 802)-Net over F7 — Constructive and digital
Digital (39, 52, 802)-net over F7, using
- net defined by OOA [i] based on linear OOA(752, 802, F7, 13, 13) (dual of [(802, 13), 10374, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(752, 4813, F7, 13) (dual of [4813, 4761, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(752, 4814, F7, 13) (dual of [4814, 4762, 14]-code), using
- trace code [i] based on linear OA(4926, 2407, F49, 13) (dual of [2407, 2381, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(4925, 2402, F49, 13) (dual of [2402, 2377, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4921, 2402, F49, 11) (dual of [2402, 2381, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,6]) ⊂ C([0,5]) [i] based on
- trace code [i] based on linear OA(4926, 2407, F49, 13) (dual of [2407, 2381, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(752, 4814, F7, 13) (dual of [4814, 4762, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(752, 4813, F7, 13) (dual of [4813, 4761, 14]-code), using
(39, 39+13, 4855)-Net over F7 — Digital
Digital (39, 52, 4855)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(752, 4855, F7, 13) (dual of [4855, 4803, 14]-code), using
- 47 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0, 1, 40 times 0) [i] based on linear OA(750, 4806, F7, 13) (dual of [4806, 4756, 14]-code), using
- trace code [i] based on linear OA(4925, 2403, F49, 13) (dual of [2403, 2378, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(4925, 2401, F49, 13) (dual of [2401, 2376, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(4923, 2401, F49, 12) (dual of [2401, 2378, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(12) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(4925, 2403, F49, 13) (dual of [2403, 2378, 14]-code), using
- 47 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0, 1, 40 times 0) [i] based on linear OA(750, 4806, F7, 13) (dual of [4806, 4756, 14]-code), using
(39, 39+13, 7610338)-Net in Base 7 — Upper bound on s
There is no (39, 52, 7610339)-net in base 7, because
- 1 times m-reduction [i] would yield (39, 51, 7610339)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 12 589260 509341 933686 569445 859092 881769 893605 > 751 [i]