Best Known (9, 14, s)-Nets in Base 49
(9, 14, 58828)-Net over F49 — Constructive and digital
Digital (9, 14, 58828)-net over F49, using
- net defined by OOA [i] based on linear OOA(4914, 58828, F49, 5, 5) (dual of [(58828, 5), 294126, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(4914, 117657, F49, 5) (dual of [117657, 117643, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(4913, 117650, F49, 5) (dual of [117650, 117637, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(497, 117650, F49, 3) (dual of [117650, 117643, 4]-code or 117650-cap in PG(6,49)), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(491, 7, F49, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(4914, 117657, F49, 5) (dual of [117657, 117643, 6]-code), using
(9, 14, 117658)-Net over F49 — Digital
Digital (9, 14, 117658)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4914, 117658, F49, 5) (dual of [117658, 117644, 6]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(4913, 117650, F49, 5) (dual of [117650, 117637, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(497, 117650, F49, 3) (dual of [117650, 117643, 4]-code or 117650-cap in PG(6,49)), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(497, 8, F49, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,49)), using
- dual of repetition code with length 8 [i]
- linear OA(491, 8, F49, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, 49, F49, 1) (dual of [49, 48, 2]-code), using
- Reed–Solomon code RS(48,49) [i]
- discarding factors / shortening the dual code based on linear OA(491, 49, F49, 1) (dual of [49, 48, 2]-code), using
- construction X4 applied to C([0,2]) ⊂ C([0,1]) [i] based on
(9, 14, large)-Net in Base 49 — Upper bound on s
There is no (9, 14, large)-net in base 49, because
- 3 times m-reduction [i] would yield (9, 11, large)-net in base 49, but