Best Known (52, 73, s)-Nets in Base 7
(52, 73, 240)-Net over F7 — Constructive and digital
Digital (52, 73, 240)-net over F7, using
- net defined by OOA [i] based on linear OOA(773, 240, F7, 21, 21) (dual of [(240, 21), 4967, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(773, 2401, F7, 21) (dual of [2401, 2328, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(773, 2402, F7, 21) (dual of [2402, 2329, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(773, 2402, F7, 21) (dual of [2402, 2329, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(773, 2401, F7, 21) (dual of [2401, 2328, 22]-code), using
(52, 73, 2095)-Net over F7 — Digital
Digital (52, 73, 2095)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(773, 2095, F7, 21) (dual of [2095, 2022, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(773, 2402, F7, 21) (dual of [2402, 2329, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2402 | 78−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(773, 2402, F7, 21) (dual of [2402, 2329, 22]-code), using
(52, 73, 917335)-Net in Base 7 — Upper bound on s
There is no (52, 73, 917336)-net in base 7, because
- 1 times m-reduction [i] would yield (52, 72, 917336)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 7 031743 799654 610559 943125 015469 860260 931158 186412 896573 947473 > 772 [i]