Best Known (16, 16+13, s)-Nets in Base 49
(16, 16+13, 402)-Net over F49 — Constructive and digital
Digital (16, 29, 402)-net over F49, using
- 491 times duplication [i] based on digital (15, 28, 402)-net over F49, using
- net defined by OOA [i] based on linear OOA(4928, 402, F49, 13, 13) (dual of [(402, 13), 5198, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(4928, 2413, F49, 13) (dual of [2413, 2385, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(4925, 2402, F49, 13) (dual of [2402, 2377, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(4917, 2402, F49, 9) (dual of [2402, 2385, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,6]) ⊂ C([0,4]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(4928, 2413, F49, 13) (dual of [2413, 2385, 14]-code), using
- net defined by OOA [i] based on linear OOA(4928, 402, F49, 13, 13) (dual of [(402, 13), 5198, 14]-NRT-code), using
(16, 16+13, 2047)-Net over F49 — Digital
Digital (16, 29, 2047)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4929, 2047, F49, 13) (dual of [2047, 2018, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4929, 2415, F49, 13) (dual of [2415, 2386, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(4925, 2401, F49, 13) (dual of [2401, 2376, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(4915, 2401, F49, 8) (dual of [2401, 2386, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(494, 14, F49, 4) (dual of [14, 10, 5]-code or 14-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(12) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(4929, 2415, F49, 13) (dual of [2415, 2386, 14]-code), using
(16, 16+13, 4814623)-Net in Base 49 — Upper bound on s
There is no (16, 29, 4814624)-net in base 49, because
- 1 times m-reduction [i] would yield (16, 28, 4814624)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 211587 792125 938772 025952 935416 374646 968619 430913 > 4928 [i]