Best Known (58, 58+15, s)-Nets in Base 5
(58, 58+15, 2232)-Net over F5 — Constructive and digital
Digital (58, 73, 2232)-net over F5, using
- net defined by OOA [i] based on linear OOA(573, 2232, F5, 15, 15) (dual of [(2232, 15), 33407, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(573, 15625, F5, 15) (dual of [15625, 15552, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(573, 15626, F5, 15) (dual of [15626, 15553, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(573, 15626, F5, 15) (dual of [15626, 15553, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(573, 15625, F5, 15) (dual of [15625, 15552, 16]-code), using
(58, 58+15, 10525)-Net over F5 — Digital
Digital (58, 73, 10525)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(573, 10525, F5, 15) (dual of [10525, 10452, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(573, 15626, F5, 15) (dual of [15626, 15553, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(573, 15626, F5, 15) (dual of [15626, 15553, 16]-code), using
(58, 58+15, large)-Net in Base 5 — Upper bound on s
There is no (58, 73, large)-net in base 5, because
- 13 times m-reduction [i] would yield (58, 60, large)-net in base 5, but