Best Known (38, 53, s)-Nets in Base 7
(38, 53, 345)-Net over F7 — Constructive and digital
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
(38, 53, 2260)-Net over F7 — Digital
Digital (38, 53, 2260)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(753, 2260, F7, 15) (dual of [2260, 2207, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(753, 2414, F7, 15) (dual of [2414, 2361, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(749, 2402, F7, 15) (dual of [2402, 2353, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(741, 2402, F7, 11) (dual of [2402, 2361, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(74, 12, F7, 3) (dual of [12, 8, 4]-code or 12-cap in PG(3,7)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(753, 2414, F7, 15) (dual of [2414, 2361, 16]-code), using
(38, 53, 1068161)-Net in Base 7 — Upper bound on s
There is no (38, 53, 1068162)-net in base 7, because
- 1 times m-reduction [i] would yield (38, 52, 1068162)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 88 125164 686080 392452 069608 184514 972645 958337 > 752 [i]