Best Known (32, 32+15, s)-Nets in Base 49
(32, 32+15, 16809)-Net over F49 — Constructive and digital
Digital (32, 47, 16809)-net over F49, using
- 491 times duplication [i] based on digital (31, 46, 16809)-net over F49, using
- net defined by OOA [i] based on linear OOA(4946, 16809, F49, 15, 15) (dual of [(16809, 15), 252089, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4946, 117664, F49, 15) (dual of [117664, 117618, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4946, 117665, F49, 15) (dual of [117665, 117619, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(4943, 117650, F49, 15) (dual of [117650, 117607, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 496−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 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(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([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4946, 117665, F49, 15) (dual of [117665, 117619, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4946, 117664, F49, 15) (dual of [117664, 117618, 16]-code), using
- net defined by OOA [i] based on linear OOA(4946, 16809, F49, 15, 15) (dual of [(16809, 15), 252089, 16]-NRT-code), using
(32, 32+15, 112937)-Net over F49 — Digital
Digital (32, 47, 112937)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4947, 112937, F49, 15) (dual of [112937, 112890, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4947, 117668, F49, 15) (dual of [117668, 117621, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(4943, 117649, F49, 15) (dual of [117649, 117606, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(4928, 117649, F49, 10) (dual of [117649, 117621, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 493−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(494, 19, F49, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,49)), using
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- Reed–Solomon code RS(45,49) [i]
- discarding factors / shortening the dual code based on linear OA(494, 49, F49, 4) (dual of [49, 45, 5]-code or 49-arc in PG(3,49)), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(4947, 117668, F49, 15) (dual of [117668, 117621, 16]-code), using
(32, 32+15, large)-Net in Base 49 — Upper bound on s
There is no (32, 47, large)-net in base 49, because
- 13 times m-reduction [i] would yield (32, 34, large)-net in base 49, but