Best Known (27, 52, s)-Nets in Base 49
(27, 52, 344)-Net over F49 — Constructive and digital
Digital (27, 52, 344)-net over F49, using
- t-expansion [i] based on digital (21, 52, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
(27, 52, 1206)-Net over F49 — Digital
Digital (27, 52, 1206)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4952, 1206, F49, 2, 25) (dual of [(1206, 2), 2360, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4952, 2412, F49, 25) (dual of [2412, 2360, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4952, 2413, F49, 25) (dual of [2413, 2361, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(4949, 2402, F49, 25) (dual of [2402, 2353, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4941, 2402, F49, 21) (dual of [2402, 2361, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4952, 2413, F49, 25) (dual of [2413, 2361, 26]-code), using
- OOA 2-folding [i] based on linear OA(4952, 2412, F49, 25) (dual of [2412, 2360, 26]-code), using
(27, 52, 1680552)-Net in Base 49 — Upper bound on s
There is no (27, 52, 1680553)-net in base 49, because
- 1 times m-reduction [i] would yield (27, 51, 1680553)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 158 490150 294957 258441 116678 654930 857012 703847 608665 120830 992357 271434 420610 764788 663873 > 4951 [i]