Best Known (54−15, 54, s)-Nets in Base 7
(54−15, 54, 345)-Net over F7 — Constructive and digital
Digital (39, 54, 345)-net over F7, using
- 71 times duplication [i] based on digital (38, 53, 345)-net over F7, using
- net defined by OOA [i] based on linear OOA(753, 345, F7, 15, 15) (dual of [(345, 15), 5122, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(753, 2416, F7, 15) (dual of [2416, 2363, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(753, 2417, F7, 15) (dual of [2417, 2364, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- linear OA(749, 2401, F7, 15) (dual of [2401, 2352, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(737, 2401, F7, 11) (dual of [2401, 2364, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(74, 16, F7, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,7)), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(753, 2417, F7, 15) (dual of [2417, 2364, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(753, 2416, F7, 15) (dual of [2416, 2363, 16]-code), using
- net defined by OOA [i] based on linear OOA(753, 345, F7, 15, 15) (dual of [(345, 15), 5122, 16]-NRT-code), using
(54−15, 54, 2443)-Net over F7 — Digital
Digital (39, 54, 2443)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(754, 2443, F7, 15) (dual of [2443, 2389, 16]-code), using
- 33 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 21 times 0) [i] based on linear OA(750, 2406, F7, 15) (dual of [2406, 2356, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(749, 2401, F7, 15) (dual of [2401, 2352, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(745, 2401, F7, 13) (dual of [2401, 2356, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(71, 5, F7, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- 33 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 21 times 0) [i] based on linear OA(750, 2406, F7, 15) (dual of [2406, 2356, 16]-code), using
(54−15, 54, 1410475)-Net in Base 7 — Upper bound on s
There is no (39, 54, 1410476)-net in base 7, because
- 1 times m-reduction [i] would yield (39, 53, 1410476)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 616 874904 011748 353820 827858 396895 259647 802273 > 753 [i]