Best Known (52−17, 52, s)-Nets in Base 49
(52−17, 52, 14708)-Net over F49 — Constructive and digital
Digital (35, 52, 14708)-net over F49, using
- net defined by OOA [i] based on linear OOA(4952, 14708, F49, 17, 17) (dual of [(14708, 17), 249984, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4952, 117665, F49, 17) (dual of [117665, 117613, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(4949, 117650, F49, 17) (dual of [117650, 117601, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(493, 15, F49, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,49) or 15-cap in PG(2,49)), using
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- Reed–Solomon code RS(46,49) [i]
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(4952, 117665, F49, 17) (dual of [117665, 117613, 18]-code), using
(52−17, 52, 74671)-Net over F49 — Digital
Digital (35, 52, 74671)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4952, 74671, F49, 17) (dual of [74671, 74619, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4952, 117665, F49, 17) (dual of [117665, 117613, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(4949, 117650, F49, 17) (dual of [117650, 117601, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(4937, 117650, F49, 13) (dual of [117650, 117613, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(493, 15, F49, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,49) or 15-cap in PG(2,49)), using
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- Reed–Solomon code RS(46,49) [i]
- discarding factors / shortening the dual code based on linear OA(493, 49, F49, 3) (dual of [49, 46, 4]-code or 49-arc in PG(2,49) or 49-cap in PG(2,49)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4952, 117665, F49, 17) (dual of [117665, 117613, 18]-code), using
(52−17, 52, large)-Net in Base 49 — Upper bound on s
There is no (35, 52, large)-net in base 49, because
- 15 times m-reduction [i] would yield (35, 37, large)-net in base 49, but