Best Known (48, 48+15, s)-Nets in Base 5
(48, 48+15, 448)-Net over F5 — Constructive and digital
Digital (48, 63, 448)-net over F5, using
- 51 times duplication [i] based on digital (47, 62, 448)-net over F5, using
- net defined by OOA [i] based on linear OOA(562, 448, F5, 15, 15) (dual of [(448, 15), 6658, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(562, 3137, F5, 15) (dual of [3137, 3075, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(561, 3126, F5, 15) (dual of [3126, 3065, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(551, 3126, F5, 13) (dual of [3126, 3075, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(562, 3137, F5, 15) (dual of [3137, 3075, 16]-code), using
- net defined by OOA [i] based on linear OOA(562, 448, F5, 15, 15) (dual of [(448, 15), 6658, 16]-NRT-code), using
(48, 48+15, 3046)-Net over F5 — Digital
Digital (48, 63, 3046)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(563, 3046, F5, 15) (dual of [3046, 2983, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(563, 3138, F5, 15) (dual of [3138, 3075, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(562, 3137, F5, 15) (dual of [3137, 3075, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(561, 3126, F5, 15) (dual of [3126, 3065, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(551, 3126, F5, 13) (dual of [3126, 3075, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(51, 11, F5, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(562, 3137, F5, 15) (dual of [3137, 3075, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(563, 3138, F5, 15) (dual of [3138, 3075, 16]-code), using
(48, 48+15, 1311397)-Net in Base 5 — Upper bound on s
There is no (48, 63, 1311398)-net in base 5, because
- 1 times m-reduction [i] would yield (48, 62, 1311398)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 21 684131 540447 902745 479174 487611 942711 062201 > 562 [i]