Best Known (27, 27+27, s)-Nets in Base 49
(27, 27+27, 344)-Net over F49 — Constructive and digital
Digital (27, 54, 344)-net over F49, using
- t-expansion [i] based on digital (21, 54, 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, 27+27, 927)-Net over F49 — Digital
Digital (27, 54, 927)-net over F49, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4954, 927, F49, 2, 27) (dual of [(927, 2), 1800, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4954, 1203, F49, 2, 27) (dual of [(1203, 2), 2352, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4954, 2406, F49, 27) (dual of [2406, 2352, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4954, 2407, F49, 27) (dual of [2407, 2353, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(4953, 2402, F49, 27) (dual of [2402, 2349, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2402 | 494−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- 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(491, 5, F49, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4954, 2407, F49, 27) (dual of [2407, 2353, 28]-code), using
- OOA 2-folding [i] based on linear OA(4954, 2406, F49, 27) (dual of [2406, 2352, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(4954, 1203, F49, 2, 27) (dual of [(1203, 2), 2352, 28]-NRT-code), using
(27, 27+27, 918250)-Net in Base 49 — Upper bound on s
There is no (27, 54, 918251)-net in base 49, because
- 1 times m-reduction [i] would yield (27, 53, 918251)-net in base 49, but
- the generalized Rao bound for nets shows that 49m ≥ 380533 289139 951918 429085 424977 620494 220759 956640 220186 603133 780867 867931 176460 790726 768849 > 4953 [i]