Best Known (18, 18+15, s)-Nets in Base 49
(18, 18+15, 344)-Net over F49 — Constructive and digital
Digital (18, 33, 344)-net over F49, using
- 492 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
(18, 18+15, 1703)-Net over F49 — Digital
Digital (18, 33, 1703)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4933, 1703, F49, 15) (dual of [1703, 1670, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4933, 2415, F49, 15) (dual of [2415, 2382, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [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(4919, 2401, F49, 10) (dual of [2401, 2382, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2400 = 492−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(14) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(4933, 2415, F49, 15) (dual of [2415, 2382, 16]-code), using
(18, 18+15, 3752214)-Net in Base 49 — Upper bound on s
There is no (18, 33, 3752215)-net in base 49, because
- 1 times m-reduction [i] would yield (18, 32, 3752215)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 1 219760 824497 625949 010024 818712 071583 807314 184261 628465 > 4932 [i]