Best Known (17, 17+15, s)-Nets in Base 49
(17, 17+15, 344)-Net over F49 — Constructive and digital
Digital (17, 32, 344)-net over F49, using
- 491 times duplication [i] based on digital (16, 31, 344)-net over F49, using
- net defined by OOA [i] based on linear OOA(4931, 344, F49, 15, 15) (dual of [(344, 15), 5129, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4931, 2409, F49, 15) (dual of [2409, 2378, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(4929, 2401, F49, 15) (dual of [2401, 2372, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(4923, 2401, F49, 12) (dual of [2401, 2378, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(492, 8, F49, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,49)), using
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- Reed–Solomon code RS(47,49) [i]
- discarding factors / shortening the dual code based on linear OA(492, 49, F49, 2) (dual of [49, 47, 3]-code or 49-arc in PG(1,49)), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(4931, 2409, F49, 15) (dual of [2409, 2378, 16]-code), using
- net defined by OOA [i] based on linear OOA(4931, 344, F49, 15, 15) (dual of [(344, 15), 5129, 16]-NRT-code), using
(17, 17+15, 1261)-Net over F49 — Digital
Digital (17, 32, 1261)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4932, 1261, F49, 15) (dual of [1261, 1229, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4932, 2413, F49, 15) (dual of [2413, 2381, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(4929, 2402, F49, 15) (dual of [2402, 2373, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(4921, 2402, F49, 11) (dual of [2402, 2381, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(493, 11, F49, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,49) or 11-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(4932, 2413, F49, 15) (dual of [2413, 2381, 16]-code), using
(17, 17+15, 2151943)-Net in Base 49 — Upper bound on s
There is no (17, 32, 2151944)-net in base 49, because
- 1 times m-reduction [i] would yield (17, 31, 2151944)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 24893 119722 445199 646600 201719 559204 234996 166828 228225 > 4931 [i]