Best Known (23, 23+11, s)-Nets in Base 49
(23, 23+11, 23532)-Net over F49 — Constructive and digital
Digital (23, 34, 23532)-net over F49, using
- net defined by OOA [i] based on linear OOA(4934, 23532, F49, 11, 11) (dual of [(23532, 11), 258818, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(4934, 117661, F49, 11) (dual of [117661, 117627, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(4934, 117665, F49, 11) (dual of [117665, 117631, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(4931, 117650, F49, 11) (dual of [117650, 117619, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(4919, 117650, F49, 7) (dual of [117650, 117631, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,5]) ⊂ C([0,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4934, 117665, F49, 11) (dual of [117665, 117631, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(4934, 117661, F49, 11) (dual of [117661, 117627, 12]-code), using
(23, 23+11, 117665)-Net over F49 — Digital
Digital (23, 34, 117665)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4934, 117665, F49, 11) (dual of [117665, 117631, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,3]) [i] based on
- linear OA(4931, 117650, F49, 11) (dual of [117650, 117619, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(4919, 117650, F49, 7) (dual of [117650, 117631, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (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([0,5]) ⊂ C([0,3]) [i] based on
(23, 23+11, large)-Net in Base 49 — Upper bound on s
There is no (23, 34, large)-net in base 49, because
- 9 times m-reduction [i] would yield (23, 25, large)-net in base 49, but